Question

Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.

32. What is the determinant of an elementary scaling matrix with k on the diagonal?

Short answer

The determinant of a \(3 \times 3\) elementary scaling matrix with \(k\) on the diagonal is \(k\).

Step-by-step solution

State the elementary matrices from Exercises 25-28

\(\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&1&0\\0&k&1\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}0&0&1\\0&1&0\\1&0&0\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&1&0\\k&0&1\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}k&0&0\\0&1&0\\0&0&1\end{aligned}} \right]\)

Determine the determinant of an elementary scaling matrix

A \(3 \times 3\) elementary scaling matrix with \(k\)on the diagonal appears as one of the three matrices shown below.

\(\left[ {\begin{aligned}{*{20}{c}}k&0&0\\0&1&0\\0&0&1\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&k&0\\0&0&1\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}1&0&0\\0&1&0\\0&0&k\end{aligned}} \right]\)

The matrix is triangular in each of the cases. It means the determinant of the matrix is the product of its diagonal entries, which is \(k\).

Therefore, a \(3 \times 3\) elementary scaling matrix with \(k\) on the diagonal has a determinant of \(k\).