Chapter 10, Problem 73RE

To determine

To find: the determinant of the given matrix

Answer to Problem 73RE

The determinant $=24$

Explanation of Solution

Given:

  $\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 2\hfill & 0\hfill & 2\hfill \\ 0\hfill & 0\hfill & 3\hfill & 3\hfill \\ 0\hfill & 0\hfill & 0\hfill & 4\hfill \end{array}\right]$

Calculation:

We begin with the $4\times 8$ matrix whose left half is $A$ and whose right half is the identity

  $\left[\begin{array}{cccccccc}1& 0& 0& 1& 1& 0& 0& 0\\ 0& 2& 0& 2& 0& 1& 0& 0\\ 0& 0& 3& 3& 0& 0& 1& 0\\ 0& 0& 0& 4& 0& 0& 0& 1\end{array}\right]$

  $\left[\begin{array}{cccccccc}1& 0& 0& 1& 1& 0& 0& 0\\ 0& 2& 0& 2& 0& 1& 0& 0\\ 0& 0& 3& 3& 0& 0& 1& 0\\ 0& 0& 0& 4& 0& 0& 0& 1\end{array}\right]$

  $\frac{R_2}{2}\to \frac{R_3}{3},\frac{R_4}{4}$

  $\left[\begin{array}{cccccccc}1& 0& 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 1/2& 0& 0\\ 0& 0& 1& 0& 0& 0& 1/3& 0\\ 0& 0& 0& 1& 0& 0& 0& 1/4\end{array}\right]$

  $R_1-R_4\to R_2-R_4$

  $\left[\begin{array}{cccccccc}1& 0& 0& 0& 1& 0& 0& -1/4\\ 0& 1& 0& 0& 0& 1/2& 0& -1/4\\ 0& 0& 1& 0& 0& 0& 1/3& 0\\ 0& 0& 0& 1& 0& 0& 0& 1/4\end{array}\right]$

  $R_3-R_4$

  $\left[\begin{array}{cccccccc}1& 0& 0& 0& 1& 0& 0& -1/4\\ 0& 1& 0& 0& 0& 1/2& 0& -1/4\\ 0& 0& 1& 0& 0& 0& 1/3& -1/4\\ 0& 0& 0& 1& 0& 0& 0& 1/4\end{array}\right]$

Then the inverse matrix of the given matrix is

  $\left[\begin{array}{cccc}1& 0& 0& -1/4\\ 0& 1/2& 0& -1/4\\ 0& 0& 1/3& -1/4\\ 0& 0& 0& 1/4\end{array}\right]$

And the determinant of $A$

  $\begin{array}{l}A=\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 2\hfill & 0\hfill & 2\hfill \\ 0\hfill & 0\hfill & 3\hfill & 3\hfill \\ 0\hfill & 0\hfill & 0\hfill & 4\hfill \end{array}\right]\\ =4(1)\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 2\hfill & 0\hfill \\ 0\hfill & 0\hfill & 3\hfill \end{array}\right]\\ =4(1)\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]\\ =4\left(1\right)\left(6\right)=24\end{array}$

Then the determinant $=24$

Conclusion:

Therefore, the determinant $=24$