###### Exercise10

In this section, we studied the effect of row operations on the matrix $$A\text{.}$$ In this exercise, we will study the effect of analogous column operations.

Suppose that $$A$$ is the $$3\times3$$ matrix $$A= \left[\begin{array}{rrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array}\right]\text{.}$$ Also consider elementary matrices

\begin{equation*} R = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ -3 \amp 0 \amp 1 \\ \end{array}\right], S = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 3 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right], P = \left[\begin{array}{rrr} 0 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \\ 1 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}
1. Explain why the matrix $$AR$$ is obtained from $$A$$ by replacing the first column $$\vvec_1$$ by $$\vvec_1 - 3\vvec_3\text{.}$$ We call this a column replacement operation. Explain why column replacement operations do not change the determinant.

2. Explain why the matrix $$AS$$ is obtained from $$A$$ by multiplying the second column by $$3\text{.}$$ Explain the effect that scaling a column has on the determinant of a matrix.

3. Explain why the matrix $$AP$$ is obtained from $$A$$ by interchanging the first and third columns. What is the effect of this operation on the determinant?

4. Use column operations to compute the determinant of

\begin{equation*} A=\left[\begin{array}{rrr} 0 \amp -3 \amp 1 \\ 1 \amp 1 \amp 4 \\ 1 \amp 1 \amp 0 \\ \end{array} \right]\text{.} \end{equation*}
in-context