(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).

(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) \(-20\)

(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).

(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) \(-2\)

(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).

(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) \(25\)

(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).

(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) \(2\)

(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).

(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) \(-8\)

(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).

(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) \(25\)

(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).

(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) \(25\)

(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).

(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) \(15\)

(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).

(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) \(-4\)

(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).

(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) \(-12\)

(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).

(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) \(9\)

(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).

(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) \(-8\)

(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).

(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) \(-25\)

(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).

(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) \(4\)

(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).

(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) \(12\)

(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).

(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) \(-2\)

(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).

(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) \(-9\)

(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).

(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) \(-9\)

(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).

(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) \(-4\)

(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).

(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) \(-3\)

(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).

(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) \(3\)

(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).

(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) \(15\)

(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).

(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) \(8\)

(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).

(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) \(5\)

(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).

(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) \(-3\)

(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).

(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) \(16\)

(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).

(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) \(4\)

(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).

(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) \(-20\)

(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).

(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) \(20\)

(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).

(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) \(-5\)

(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).

(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) \(4\)

(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).

(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) \(-12\)

(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).

(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) \(-2\)

(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).

(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) \(-10\)

(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).

(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) \(-10\)

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

(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).

(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) \(4\)

(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).

(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) \(-3\)

(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).

(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) \(-4\)

(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).

(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) \(-6\)

(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).

(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) \(-12\)

(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).

(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) \(-2\)

(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).

(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) \(3\)

(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).

(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) \(-8\)

(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).

(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) \(-10\)

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

(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).

(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) \(-8\)

(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).

(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) \(25\)

(a) Give a \(3\times 3\) matrix \(P\) 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 \({R_3} \leftrightarrow {R_1}\) to \(A\) (note the order).

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

(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) \(9\)

(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).

(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) \(8\)

(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).

(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) \(-10\)

(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).

(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) \(-3\)