Here,

To show:

We have to find A^-1 and

Firstly, we find the adj A and for that we have to find co-factors:

a₁₁ (co – factor of 2) = (-1)¹⁺¹(-2) = (-1)²(-2) = -2

a₁₂ (co – factor of 3) = (-1)¹⁺²(5) = (-1)³(5) = -5

a₂₁ (co – factor of 5) = (-1)²⁺¹(3) = (-1)³(3) = -3

a₂₂ (co – factor of -2) = (-1)²⁺²(2) = (-1)⁴(2) = 2

Now, adj A = Transpose of co-factor Matrix

Calculating |A|

= [2 × (-2) – 3 × 5]

= (-4 – 15)

= -19

[Taking (-1) common from the matrix]

Hence Proved