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*}
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