Explanation:
Let's calculate A(adj A) and then show that it is equal to IAI, where I is the identity matrix.
First, let's find the adjoint (adjugate) of matrix A.
Given matrix A:
A = [1 2]
[3 4]
The adjoint of A, denoted as adj A, is the transpose of the cofactor matrix of A.
Cofactor of element a_ij = (-1)^(i+j) * det(minor_ij)
where minor_ij is the (i, j) minor obtained by removing the i-th row and j-th column.
For A = [1 2]
[3 4]
Cofactor of a_11 (element 1): (-1)^(1+1) * det(minor_11) = 1 * (4) = 4
Cofactor of a_12 (element 2): (-1)^(1+2) * det(minor_12) = -1 * (3) = -3
Cofactor of a_21 (element 3): (-1)^(2+1) * det(minor_21) = -1 * (2) = -2
Cofactor of a_22 (element 4): (-1)^(2+2) * det(minor_22) = 1 * (1) = 1
Now, we construct the adjugate matrix (adj A) by taking the transpose of the cofactor matrix:
adj A = [4 -2]
[-3 1]
Next, let's calculate the product A(adj A):
A(adj A) = [1 2] * [4 -2] = [1*4 + 2*(-3) 1*(-2) + 2*1] = [4 -2]
[3 4] [-3*4 + 4*(-2) -3*(-2) + 4*1] [0 10]
Now, let's find the magnitude of matrix A (|A|) and the identity matrix I:
|A| = |[1 2]|
|[3 4]| = (1*4) - (2*3) = 4 - 6 = -2
The identity matrix I of size 2x2 is:
I = [1 0]
[0 1]
Finally, let's calculate I * |A|:
I * |A| = [1 0] * (-2) = [-2 0]
[0 1] [0 -2]
Comparing A(adj A) and I * |A|:
A(adj A) = [4 -2]
[0 10]
I * |A| = [-2 0]
[0 -2]
We can see that A(adj A) is equal to I * |A|, which means that A(adj A) = I * |A| holds true.