To find the inverse of a matrix A, we need to check if the determinant of A is zero. If the determinant is zero, the inverse does not exist.

Finding the determinant of matrix A:
The given matrix A is a 3 × 3 matrix with elements given by aij = i^2 – j^2, where 1 ≤ i, j ≤ 3.

To find the determinant, we can use the formula for a 3 × 3 matrix:

det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

Substituting the given values of aij into the formula:

det(A) = (1^2 - 1^2)(2^2 - 3^2) - (1^2 - 2^2)(1^2 - 3^2) + (1^2 - 3^2)(1^2 - 2^2)
= (0)(-5) - (-3)(-8) + (-8)(-3)
= 0 + 24 + 24
= 48

Since the determinant of A is 48, which is not equal to zero, the inverse of matrix A exists.

Finding the inverse of matrix A:
The inverse of a 3 × 3 matrix A is given by the formula:

A^(-1) = (1/det(A)) * adj(A)

where adj(A) represents the adjugate of matrix A.

To find the adjugate of matrix A, we need to find the cofactor matrix of matrix A and then take its transpose.

Cofactor matrix of matrix A:
The cofactor of each element aij is given by (-1)^(i+j) * Mij, where Mij is the determinant of the submatrix obtained by deleting the row and column containing aij.

For each element of A, we can calculate the corresponding cofactor:

C11 = (-1)^(1+1) * det([a22 a23; a32 a33]) = (-1) * ((2^2 - 3^2)(1^2 - 3^2)) = (-1)(-5) = 5
C12 = (-1)^(1+2) * det([a21 a23; a31 a33]) = (1) * ((1^2 - 3^2)(1^2 - 3^2)) = (-8)(-8) = 64
C13 = (-1)^(1+3) * det([a21 a22; a31 a32]) = (-1) * ((1^2 - 2^2)(1^2 - 2^2)) = (-3)(-3) = 9
C21 = (-1)^(2+1) * det([a12 a13; a32 a33]) = (1) * ((0)(1^2 - 3^2)) = 0
C22 = (-1)^(2+2) * det([a11 a13; a31 a33]) = (-1) * ((0)(1^2 - 3^2)) = 0
C23 = (-1)^(2+3) *

Aij = i2 – j2∀i, j