Find the inverse of the matrix . Hence, find the matrix P satisfying the matrix equation .

Let the given matrix we have to find A^-1.

Now A^-1 will exist only of |A| ≠ 0

Let us first find |A|

⇒ |A| = (-3) (-3) – (5)(2)

⇒ |A| = 9 – 10 = -1

Hence |A| ≠ 0 and A^-1 exist

Now A^-1 is given by

Now adjoint(A) = C^T where C is the cofactor matrix and C^T is the transpose of cofactor matrix

The cofactor is given by

⇒ C_ij = (-1) ^i+jM_ij …(a)

Where M represents minor

M_ij is the determinant of matrix leaving the i^th row and j^th column

The cofactor matrix C will be

Now transpose of C that is C^T

For C^T we will interchange the rows and columns

Using (a)

From matrix A the minors are

M₁₁ = -3, M₁₂ = 5, M₂₁ = 2 and M₂₂ = -3

Now inverse of A is

Now we have to solve the matrix equation

As we know that AA^-1 = I hence multiply equation by A^-1 from right side where I is identity matrix

Hence