Chapter 8.4, Problem 73E

To determine

To calculate: The matrix with the conditions dimension as $2\times 2$ , $\left|A\right|=0$ and $A\ne O$ .

Answer to Problem 73E

The matrix with the conditions dimension as $2\times 2$ , $\left|A\right|=0$ and $A\ne O$ is $\begin{array}{cc}2& 2\\ 3& 3\end{array}$ .

Explanation of Solution

Given information:

The conditions dimensionas $2\times 2$ , $\left|A\right|=0$ and $A\ne O$ .

Formula used:

For the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Calculation:

Consider the conditions dimensionas $2\times 2$ , $\left|A\right|=0$ and $A\ne O$ .

One of such matrices can be $\begin{array}{cc}2& 2\\ 3& 3\end{array}$ . Its dimension is $2\times 2$ . It is a nonzero matrix.

Also, recall that for matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

Therefore, determinant is,

  $\begin{array}{c}\begin{array}{cc}2& 2\\ 3& 3\end{array}=2\left(3\right)-2\left(3\right)\\ =6-6\\ =0\end{array}$

That is, $\left|A\right|=0$ .

Thus, the matrix with the conditions, dimension as $2\times 2$ , $\left|A\right|=0$ and $A\ne O$ is $\begin{array}{cc}2& 2\\ 3& 3\end{array}$ .