Chapter 3.8, Problem 40HP

To determine

To state : whether the given statement is always, sometimes or never true.

Answer to Problem 40HP

The given statement is sometime true.

Explanation of Solution

Given information :

Given that a square matrix has a multiplicative inverse.

When the determinant of a square matrix is zero then the multiplicative inverse doesn’t exist.

Therefore, the given statement is sometime true.

Consider an example to justify the answer.

  $A=\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$

Its $\left|A\right|=6-5=1$

Therefore, its inverse is

  $\begin{array}{c}{A}^{-1}=\frac{1}{\left|A\right|}adj\left(A\right)\\ =\left[\begin{array}{cc}2& -1\\ -5& 3\end{array}\right]\end{array}$

Also there doesn’t exist multiplicative inverse of the matrix $B=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$ .