Chapter 3.7, Problem 59HP

To determine

To solve:The determinant of 3X3 matrix with the given equation.

Answer to Problem 59HP

The determinant of 3X3 matrix is 0.

Explanation of Solution

Given information:

The system of equation is

  $a_{mn}=\{\begin{array}{l}0\text{ if }m+n\text{ is even}\\ m+n\text{ if }m+n\text{ is odd }\end{array}$

Calculation:

  $\begin{array}{l}\text{Consider the matrix defined as follows: }\\ a_{mn}=\{\begin{array}{l}0\text{ if }m+n\text{ is even}\\ m+n\text{ if }m+n\text{ is odd }\end{array}\\ \end{array}$

The matrix has values as follows,

  $A=\left[\begin{array}{ccc}0& 3& 0\\ 3& 0& 5\\ 0& 5& 0\end{array}\right]$

The determinant of the matrix is:

  $\begin{array}{l}\left|A\right|=\left|\begin{array}{ccc}0& 3& 0\\ 3& 0& 5\\ 0& 5& 0\end{array}\right|\\ =0\left|\begin{array}{cc}0& 5\\ 5& 0\end{array}\right|+3\left|\begin{array}{cc}5& 3\\ 0& 0\end{array}\right|+0\left|\begin{array}{cc}3& 0\\ 0& 5\end{array}\right|\\ =0+3\times 0+0\\ =0\end{array}$

Hence the value of the determinant is 0.