Q10. - Algebra 2

Question

Determine whether each pair of matrices are inverses.10.

$\mathbf{P}\mathbf{=}\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$

$\mathbf{Q}\mathbf{=}\left[\begin{array}{cc}-1& 1\\ 1& 0\end{array}\right]$

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 matrix A 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:

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

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.