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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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)` .

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