Question

Suppose that A is a square n by n matrix that is nonsingular.

The matrix B such that AB = BA = In is called the__________

of the matrix A.

Short answer

Inverse.

Step-by-step solution

Step 1. Given Information

A square matrix A which is nonsingular also

$AB=BA=I_n$

Step 2. Calculation

$$\begin{gathered} Consider, \\ A=\left[\begin{array}{cc}3& 1\\ 1& -2\end{array}\right] \\ B=\left[\begin{array}{cc}\frac{2}{7}& \frac{1}{7}\\ \frac{1}{7}& -\frac{3}{7}\end{array}\right] \\ AB=\left[\begin{array}{cc}3& 1\\ 1& -2\end{array}\right]\left[\begin{array}{cc}\frac{2}{7}& \frac{1}{7}\\ \frac{1}{7}& -\frac{3}{7}\end{array}\right] \\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$

thus,$A{A}^{-1}=I_n$