A is a square matrix. Write the value of A(adj A)

To find the adjugate of a square matrix A, we'll first
find the cofactor matrix of A.

The cofactor (i,j) of the
cofactor matrix of A is:

`C_(i,j) = (-1)^(i+j)*M_(i,j)` ,
where `M_(i,j)` represents a minor of the matrix A which is the determinant that can be
found suppressing the row i and the column j of the matrix
A.

Let's calculate the adj.(A), where A is 2*2 square
matrix:

(a , b)

A
=

(c , d)

We'll calculate the
cofactor elements of the cofactor matrix A.

`C_(1,1) =
(-1)^(1+1)*M_(1,1,)`

We notice that if we'll suppress the
1st row and the 1st column, we'll get the element d, therefore the minor `M_(1,1)` is
the element d.

`M_(1,1) =
d`

`C_(1,1) = d`

`C_(1,2) =
(-1)^(3)*M_(1,2)`

We notice that if we'll
suppress the 1st row and the 2nd column, we'll get the element
c.

`C_(1,2) = -c`

`C_(2,1) =
(-1)^(3)*b`

`C_(2,1) = -
b`

`C_(2,2) = a`

The cofactor
matrix of A is:

(d , -c)

C
=

(-b , a)

Now, we'll
calculate the transpose of the cofactor matrix, such as the 1st row (d , -c) becomes the
1st column and the 2nd row (-b , a) becoms the 2nd
column:

(d , -b)

`C^(T) =
`

(-c , a).

The
adjoint of the matrix A is the transpose of the cofactor matrix: adj.(A) = `C^(T)`
.