## 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).

(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) $$-20$$

## 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).

(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) $$-2$$

## 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).

(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) $$25$$

## 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).

(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) $$2$$

## 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).

(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) $$-8$$

## 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).

(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) $$25$$

## 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).

(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) $$25$$

## 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).

(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) $$15$$

## 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).

(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) $$-4$$

## 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).

(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) $$-12$$

## 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).

(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) $$9$$

## 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).

(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) $$-8$$

## 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).

(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) $$-25$$

## 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).

(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) $$4$$

## 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).

(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) $$12$$

## 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).

(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) $$-2$$

## 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).

(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) $$-9$$

## 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).

(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) $$-9$$

## 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).

(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) $$-4$$

## 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).

(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) $$-3$$

## 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).

(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) $$3$$

## 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).

(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) $$15$$

## 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).

(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) $$8$$

## 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).

(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) $$5$$

## 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).

(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) $$-3$$

## 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).

(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) $$16$$

## 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).

(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) $$4$$

## 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).

(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) $$-20$$

## 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).

(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) $$20$$

## 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).

(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) $$-5$$

## 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).

(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) $$4$$

## 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).

(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) $$-12$$

## 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).

(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) $$-2$$

## 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).

(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) $$-10$$

## 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).

(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) $$-10$$

## 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).

(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) $$4$$

## 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).

(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) $$-3$$

## 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).

(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) $$-4$$

## 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).

(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) $$-6$$

## 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).

(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) $$-12$$

## 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).

(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) $$-2$$

## 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).

(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) $$3$$

## 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).

(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) $$-8$$

## 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).

(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) $$-10$$

## 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).

(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) $$-8$$

## 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).

(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) $$25$$

## 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).

(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) $$9$$

## 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).

(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) $$8$$

## 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).

(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) $$-10$$

## 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).

(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) $$-3$$