G1 - Row operations and matrices


Example 1

G1 - Row operations and matrices (ver. 1)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-2 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-4 \, {R_3} \to {R_3}\) and then \(-2 \, {R_1} + {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -4 \end{array}\right]\)

(c) \(PQA\)

(d) \(-20\)


Example 2

G1 - Row operations and matrices (ver. 2)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} - 4 \, {R_3} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_2} \leftrightarrow {R_1}\) and then \({R_2} - 4 \, {R_3} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -4 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-2\)


Example 3

G1 - Row operations and matrices (ver. 3)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-5\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_2} \to {R_2}\) and then \({R_2} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(25\)


Example 4

G1 - Row operations and matrices (ver. 4)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 3 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-2\). Use matrix multiplication to describe the matrix obtained by applying \({R_3} \leftrightarrow {R_2}\) and then \({R_1} + 3 \, {R_2} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(c) \(PQA\)

(d) \(2\)


Example 5

G1 - Row operations and matrices (ver. 5)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 5 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} + 5 \, {R_2} \to {R_1}\) and then \(-4 \, {R_1} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} -4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-8\)


Example 6

G1 - Row operations and matrices (ver. 6)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-5 \, {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-5 \, {R_3} \to {R_3}\) and then \({R_2} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -5 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(25\)


Example 7

G1 - Row operations and matrices (ver. 7)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-5\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)

(c) \(PQA\)

(d) \(25\)


Example 8

G1 - Row operations and matrices (ver. 8)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(4 \, {R_2} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_3} \to {R_3}\) and then \(4 \, {R_2} + {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 4 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)

(c) \(PQA\)

(d) \(15\)


Example 9

G1 - Row operations and matrices (ver. 9)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} + 4 \, {R_3} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \({R_2} + 4 \, {R_3} \to {R_2}\) and then \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(-4\)


Example 10

G1 - Row operations and matrices (ver. 10)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} - 3 \, {R_3} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-3 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} - 3 \, {R_3} \to {R_1}\) and then \(-3 \, {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3 \end{array}\right]\)

(c) \(QPA\)

(d) \(-12\)


Example 11

G1 - Row operations and matrices (ver. 11)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(2 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-3 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-3\). Use matrix multiplication to describe the matrix obtained by applying \(2 \, {R_1} + {R_3} \to {R_3}\) and then \(-3 \, {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(9\)


Example 12

G1 - Row operations and matrices (ver. 12)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 5 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} + 5 \, {R_2} \to {R_1}\) and then \(-4 \, {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-8\)


Example 13

G1 - Row operations and matrices (ver. 13)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-5 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-5\). Use matrix multiplication to describe the matrix obtained by applying \(-5 \, {R_1} \to {R_1}\) and then \({R_1} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} -5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-25\)


Example 14

G1 - Row operations and matrices (ver. 14)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(2 \, {R_1} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} + 2 \, {R_2} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \(2 \, {R_1} \to {R_1}\) and then \({R_1} + 2 \, {R_2} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(4\)


Example 15

G1 - Row operations and matrices (ver. 15)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-4 \, {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-3 \, {R_1} + {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-3\). Use matrix multiplication to describe the matrix obtained by applying \(-4 \, {R_3} \to {R_3}\) and then \(-3 \, {R_1} + {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -4 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(12\)


Example 16

G1 - Row operations and matrices (ver. 16)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} - 3 \, {R_3} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_3} \leftrightarrow {R_1}\) and then \({R_1} - 3 \, {R_3} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(PQA\)

(d) \(-2\)


Example 17

G1 - Row operations and matrices (ver. 17)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-3 \, {R_2} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(2 \, {R_1} + {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(2 \, {R_1} + {R_2} \to {R_2}\) and then \(-3 \, {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-9\)


Example 18

G1 - Row operations and matrices (ver. 18)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(3 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(3 \, {R_3} \to {R_3}\) and then \({R_2} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array}\right]\)

(c) \(PQA\)

(d) \(-9\)


Example 19

G1 - Row operations and matrices (ver. 19)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} + 3 \, {R_2} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} \leftrightarrow {R_2}\) and then \({R_1} + 3 \, {R_2} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-4\)


Example 20

G1 - Row operations and matrices (ver. 20)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} + 5 \, {R_2} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} + 5 \, {R_2} \to {R_1}\) and then \({R_2} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 5 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-3\)


Example 21

G1 - Row operations and matrices (ver. 21)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} - 3 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-3\). Use matrix multiplication to describe the matrix obtained by applying \({R_3} \leftrightarrow {R_2}\) and then \({R_1} - 3 \, {R_2} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & -3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(c) \(PQA\)

(d) \(3\)


Example 22

G1 - Row operations and matrices (ver. 22)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-3 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-3 \, {R_1} \to {R_1}\) and then \({R_1} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} -3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(15\)


Example 23

G1 - Row operations and matrices (ver. 23)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-2 \, {R_1} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-5 \, {R_1} + {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-4\). Use matrix multiplication to describe the matrix obtained by applying \(-2 \, {R_1} \to {R_1}\) and then \(-5 \, {R_1} + {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ -5 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(8\)


Example 24

G1 - Row operations and matrices (ver. 24)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_1} + {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-5\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_1} + {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 5 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(5\)


Example 25

G1 - Row operations and matrices (ver. 25)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-3 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(-3 \, {R_1} + {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(-3\)


Example 26

G1 - Row operations and matrices (ver. 26)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \(-4 \, {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -4 \end{array}\right]\)

(c) \(PQA\)

(d) \(16\)


Example 27

G1 - Row operations and matrices (ver. 27)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} - 3 \, {R_3} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-4\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} \leftrightarrow {R_3}\) and then \({R_1} - 3 \, {R_3} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(4\)


Example 28

G1 - Row operations and matrices (ver. 28)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} + 2 \, {R_3} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-4\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_2} \to {R_2}\) and then \({R_2} + 2 \, {R_3} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-20\)


Example 29

G1 - Row operations and matrices (ver. 29)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(5 \, {R_1} + {R_2} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-5\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_1} + {R_2} \to {R_2}\) and then \(-4 \, {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 5 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -4 \end{array}\right]\)

(c) \(QPA\)

(d) \(20\)


Example 30

G1 - Row operations and matrices (ver. 30)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-2 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-2 \, {R_1} + {R_3} \to {R_3}\) and then \({R_2} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(-5\)


Example 31

G1 - Row operations and matrices (ver. 31)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 2 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-4\). Use matrix multiplication to describe the matrix obtained by applying \({R_2} \leftrightarrow {R_1}\) and then \({R_1} + 2 \, {R_2} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(4\)


Example 32

G1 - Row operations and matrices (ver. 32)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(4 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(4 \, {R_2} \to {R_2}\) and then \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-12\)


Example 33

G1 - Row operations and matrices (ver. 33)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} - 2 \, {R_3} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_3} \leftrightarrow {R_1}\) and then \({R_2} - 2 \, {R_3} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-2\)


Example 34

G1 - Row operations and matrices (ver. 34)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-5 \, {R_2} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_2} \leftrightarrow {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-2\). Use matrix multiplication to describe the matrix obtained by applying \({R_2} \leftrightarrow {R_3}\) and then \(-5 \, {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(c) \(PQA\)

(d) \(-10\)


Example 35

G1 - Row operations and matrices (ver. 35)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_1} \to {R_1}\) and then \({R_3} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-10\)


Example 36

G1 - Row operations and matrices (ver. 36)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(4 \, {R_2} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-4\). Use matrix multiplication to describe the matrix obtained by applying \(4 \, {R_2} + {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 4 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(4\)


Example 37

G1 - Row operations and matrices (ver. 37)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(4 \, {R_1} + {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} \leftrightarrow {R_3}\) and then \(4 \, {R_1} + {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-3\)


Example 38

G1 - Row operations and matrices (ver. 38)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_1} + {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \({R_3} \leftrightarrow {R_1}\) and then \(-4 \, {R_1} + {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -4 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-4\)


Example 39

G1 - Row operations and matrices (ver. 39)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(2 \, {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} + 3 \, {R_3} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-3\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} + 3 \, {R_3} \to {R_1}\) and then \(2 \, {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-6\)


Example 40

G1 - Row operations and matrices (ver. 40)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 4 \, {R_3} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-3 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(4\). Use matrix multiplication to describe the matrix obtained by applying \(-3 \, {R_2} \to {R_2}\) and then \({R_1} + 4 \, {R_3} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-12\)


Example 41

G1 - Row operations and matrices (ver. 41)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-4 \, {R_2} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \(-4 \, {R_2} + {R_3} \to {R_3}\) and then \({R_1} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -4 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(-2\)


Example 42

G1 - Row operations and matrices (ver. 42)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(5 \, {R_1} + {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(-3\). Use matrix multiplication to describe the matrix obtained by applying \(5 \, {R_1} + {R_3} \to {R_3}\) and then \({R_1} \leftrightarrow {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 5 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(3\)


Example 43

G1 - Row operations and matrices (ver. 43)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} - 2 \, {R_3} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_2} - 2 \, {R_3} \to {R_2}\) and then \(-4 \, {R_1} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} -4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-8\)


Example 44

G1 - Row operations and matrices (ver. 44)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} + 4 \, {R_3} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-2 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} + 4 \, {R_3} \to {R_1}\) and then \(-2 \, {R_1} \to {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-10\)


Example 45

G1 - Row operations and matrices (ver. 45)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_1} - 3 \, {R_2} \to {R_1}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-4 \, {R_3} \to {R_3}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} - 3 \, {R_2} \to {R_1}\) and then \(-4 \, {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & -3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -4 \end{array}\right]\)

(c) \(QPA\)

(d) \(-8\)


Example 46

G1 - Row operations and matrices (ver. 46)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-5 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-5 \, {R_2} \to {R_2}\) and then \({R_3} \leftrightarrow {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(25\)


Example 47

G1 - Row operations and matrices (ver. 47)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-3 \, {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_3} \leftrightarrow {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \(-3 \, {R_3} \to {R_3}\) and then \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)

(c) \(QPA\)

(d) \(9\)


Example 48

G1 - Row operations and matrices (ver. 48)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(3 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(4 \, {R_1} \to {R_1}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(2\). Use matrix multiplication to describe the matrix obtained by applying \(4 \, {R_1} \to {R_1}\) and then \(3 \, {R_1} + {R_3} \to {R_3}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(8\)


Example 49

G1 - Row operations and matrices (ver. 49)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \(-4 \, {R_1} + {R_3} \to {R_3}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \(-2 \, {R_2} \to {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(5\). Use matrix multiplication to describe the matrix obtained by applying \(-4 \, {R_1} + {R_3} \to {R_3}\) and then \(-2 \, {R_2} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -4 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(QPA\)

(d) \(-10\)


Example 50

G1 - Row operations and matrices (ver. 50)

(a) Give a \(3\times 3\) matrix \(P\) that may be used to perform the row operation \({R_2} + 3 \, {R_3} \to {R_2}\).

(b) Give a \(3\times 3\) matrix \(Q\) that may be used to perform the row operation \({R_1} \leftrightarrow {R_2}\).

(c) Suppose \(A\) is a \(3\times 3\) matrix with determinant \(3\). Use matrix multiplication to describe the matrix obtained by applying \({R_1} \leftrightarrow {R_2}\) and then \({R_2} + 3 \, {R_3} \to {R_2}\) to \(A\) (note the order).

(d) Finally, explain how to find the determinant of the matrix described in (c).

Answer.

(a) \(P=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right]\)

(b) \(Q=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)

(c) \(PQA\)

(d) \(-3\)