Q12. - Algebra 2

Question

Determine whether each pair of matrices are inverses.12.

$\mathbf{A}\mathbf{=}\left[\begin{array}{cc}6& 2\\ 5& 2\end{array}\right]$

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

Step-by-Step Solution

Answer

The given matrices are not 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$\mathbf{AB}\mathbf{=}\mathbf{BA}\mathbf{=}\mathbf{I}$ whereI is an identity matrix of the same order as that of matrix Aor B.

Step 2 ­- Find the product of the given matrices. A ⋅ B = [ 6 2 5 2 ] ⋅ [ 1 1 − 5 2 − 3 ] = [ ( 6 ) ( 1 ) + ( 2 ) ( − 5 2 ) ( 6 ) ( 1 ) + ( 2 ) ( − 3 ) ( 5 ) ( 1 ) + ( 2 ) ( − 5 2 ) ( 5 ) ( 1 ) + ( 2 ) ( − 3 ) ] = [ 6 + ( − 5 ) 6 + ( − 6 ) 5 + ( − 5 ) 5 − 6 ] = [ 6 − 5 6 − 6 5 − 5 5 − 6 ] = [ 1 0 0 − 1 ] As the product of the given matrices is not equal to the identity matrix, therefore the given matrices are not inverse of each other.

The product of the given matrices is:

Step 3 ­- Write the conclusion.

The given matrices are not inverse of each other.