Question

Determine whether each pair of matrices are inverses.

$\mathbf{X}\mathbf{=}\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$,$\mathbf{Y}\mathbf{=}\left[\begin{array}{cc}2& -1\\ -5& 3\end{array}\right]$

Short answer

The given matrices are inverse of each other.

Step-by-step solution

­- Definition of inverse of matrix.

A square matrix $\mathbf{B}$is said to be an inverse of the square matrix $\mathbf{A}$if $\mathbf{AB}\mathbf{=}\mathbf{BA}\mathbf{=}\mathbf{I}$where $\mathbf{I}$is an identity matrix of the same order as that of matrix $\mathbf{A}$or$\mathbf{B}$.

­- Find the product of the given matrices.

The product of the given matrices is:

$\begin{array}{c}X\cdot Y=\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]\cdot \left[\begin{array}{cc}2& -1\\ -5& 3\end{array}\right]\\ =\left[\begin{array}{cc}\left(3\right)\left(2\right)+\left(1\right)\left(-5\right)& \left(3\right)\left(-1\right)+\left(1\right)\left(3\right)\\ \left(5\right)\left(2\right)+\left(2\right)\left(-5\right)& \left(5\right)\left(-1\right)+\left(2\right)\left(3\right)\end{array}\right]\\ =\left[\begin{array}{cc}6-5& -3+3\\ 10-10& -5+6\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

$\begin{array}{c}Y\cdot X=\left[\begin{array}{cc}2& -1\\ -5& 3\end{array}\right]\cdot \left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]\\ =\left[\begin{array}{cc}\left(2\right)\left(3\right)+\left(-1\right)\left(5\right)& \left(2\right)\left(1\right)+\left(-1\right)\left(2\right)\\ \left(-5\right)\left(3\right)+\left(3\right)\left(5\right)& \left(-5\right)\left(1\right)+\left(3\right)\left(2\right)\end{array}\right]\\ =\left[\begin{array}{cc}6-5& 2-2\\ -15+15& -5+6\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

As the product of the given matrices is equal to the identity matrix, therefore the given matrices are inverse of each other.

­- Write the conclusion. 

The given matrices are inverse of each other.