Chapter 4.6, Problem 40HP

To determine

To verify: whether a statement “A square matrix has a multiplicative inverse.” is sometimes , always, or never true.

Answer to Problem 40HP

Sometimes true.

Explanation of Solution

Given information:

A square matrix has a multiplicative inverse.

Formula used:

The value of a second order determinant is the difference of the products of the products of the two diagonals.

  $\det \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

Inverse of a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ is, where $ad-bc\ne 0$ .

Proof:

According to the question ,

A square matrix has a multiplicative inverse if determinant is not equal to zero.

Hence, a square matrix sometimes has a multiplicative inverse.