Question 15.16: Find the inverse matrix A^−1 for matrix A = [20 5 6 2]...

Because there are only four elements, the cofactor corresponding to each element of A will just be the element in the opposite corner, with the sign (-1)^{i  +  j} . Therefore, the corresponding cofactor matrix will be

C =  \begin{bmatrix}2&-6\\-5&20 \end{bmatrix}

The adjoint is the transpose of the cofactor matrix and so

AdjA = \begin{bmatrix}2&-5\\-6&20 \end{bmatrix}

The determinant of the original matrix A is easily calculated as

|A| = 20 × 2 − 5 × 6 = 40 − 30 = 10

The inverse matrix is thus

A^{-1}  =  \frac{AdjA}{\left|A\right| }  =  \frac{\begin{bmatrix}2&-5\\-6&20 \end{bmatrix}}{10}  =  \begin{bmatrix} 0.2 &−0.5 \\ −0.6& 2\end{bmatrix}