find the inverse of the following matrix......

Let the matrix be A
det A = 2(3) - (-1)(-5) = 1
adj A =
[ 2 .... -1 ]
[ -5 .... 3 ]
inverse of A = adj A / det A =
[ 2 .... -1 ]
[ -5 .... 3 ]

* If you don't believe, you can multiply it by yourself to check my answer.

BUT I think the anwser you claim is wrong, and adjoint or adjugate of matrix is actually a mean for computation. No one really need to understand. However, in definition, adjugate means the transpose of cofactor matrix, in which this concept arises because the senior observes that to solve a system of linear equations, the problem is just a "game of numbers", they formulate the algorithm, and this is the philosophy behind. I hope I can answer your question.

Let A be the matrix, and Aij be the element

Det (A) = 3(2) - 1(5) = 1

Minor of A11 = 2
Minor of A12 = 5
Minor of A21 = 1
Minor of A22 = 3

Cofactor Cij = (-1)^i+j x Minor
C11 = (-1)^(1+1) x A11 = 2
C12 = (-1)^(1+2) x A12 = -5
C21 = (-1)^(2+1) x A21 = -1
C22 = (-1)^(2+2) x A22 = 3

Adj (A) = Transpose of Cofactor
then,
1st row: 2 -1
2nd row: -5 3

Inverse of A = Adj(A) / Det(A)
then the answer is
[ 2 -1]
[-5 3 ]