Given:- The matrix:
[tex]\begin{bmatrix}{2} & {9} \\ {3} & {13}\end{bmatrix},\begin{bmatrix}{-13} & {9} \\ {3} & {-2}\end{bmatrix}[/tex]
To check the given matrix are inverse.
Solution:-
As we know that when the two matrices are inverse then their resultant is equal to the identity matrix I.
So, multiplying the given matrices as:
[tex]\begin{gathered} \begin{bmatrix}{2} & {9} \\ {3} & {13}\end{bmatrix}\begin{bmatrix}{-13} & {9} \\ {3} & {-2}\end{bmatrix}=\begin{bmatrix}{2\times(-13)}+9\times3 & {2\times9}+9\times(-2) \\ {3\times(-13)+13\times3} & {3\times9+13\times(-2)}\end{bmatrix} \\ \begin{bmatrix}{2} & {9} \\ {3} & {13}\end{bmatrix}\begin{bmatrix}{-13} & {9} \\ {3} & {-2}\end{bmatrix}=\begin{bmatrix}{-26+27} & {18-18} \\ {-39+39} & {27-26}\end{bmatrix} \\ \begin{bmatrix}{2} & {9} \\ {3} & {13}\end{bmatrix}\begin{bmatrix}{-13} & {9} \\ {3} & {-2}\end{bmatrix}=\begin{bmatrix}{1} & {0} \\ {0} & {1}\end{bmatrix} \end{gathered}[/tex]
As we can observe that the resultant of the two matrix multiplication is an Identity matrix.
So, the given matrices are inverse.
Final answer:-
Therefore, the given matrices are inverse of each other.
Option (C) is correct.
