Exercise 10

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

Suppose that $A$ is the $3\times3$ matrix $A= \left[\begin{array}{rrr} \vvec_1 & \vvec_2 & \vvec_3 \end{array}\right]$. Also consider elementary matrices $$R = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & 0 & 1 \\ \end{array}\right],~~~ S = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \\ \end{array}\right],~~~ P = \left[\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array}\right]$$.

1. Explain why the matrix $AR$ is obtained from $A$ by replacing the first column $\vvec_1$ by $\vvec_1 - 3\vvec_3$. 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 $3$. 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 $$A=\left[\begin{array}{rrr} 0 & -3 & 1 \\ 1 & 1 & 4 \\ 1 & 1 & 0 \\ \end{array} \right]$$.