Q11. - Algebra 2

Question

Determine whether each pair of matrices are inverses.11.

$\mathbf{R}\mathbf{=}\left[\begin{array}{cc}2& 2\\ 3& 4\end{array}\right]$

$\mathbf{S}\mathbf{=}\left[\begin{array}{cc}2& -1\\ \frac{-3}{2}& 1\end{array}\right]$

Step-by-Step Solution

Answer

The given matrices are inverse of each other.

Step 1 ­- Definition of inverse of matrix.

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

Step 2 ­- Find the product of the given matrices.

The product of the given matrices is:

$\begin{array}{c}\left[\begin{array}{cc}2& 2\\ 3& 4\end{array}\right]\cdot \left[\begin{array}{cc}2& -1\\ \frac{-3}{2}& 1\end{array}\right]=\left[\begin{array}{cc}\left(2\right)\left(2\right)+\left(2\right)\left(\frac{-3}{2}\right)& \left(2\right)(-1)+\left(2\right)\left(1\right)\\ \left(3\right)\left(2\right)+\left(4\right)\left(\frac{-3}{2}\right)& \left(3\right)(-1)+\left(4\right)\left(1\right)\end{array}\right]\\ =\left[\begin{array}{cc}4+(-3)& -2+2\\ 6+(-6)& -3+4\end{array}\right]\\ =\left[\begin{array}{cc}4-3& -2+2\\ 6-6& -3+4\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\end{array}$

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

Step 3 ­- Write the conclusion.

The given matrices are inverse of each other.