If A = [a_ ij ]_ 3 3 , where a_ ij = cases 1, & if i+j is even 0, & if i+j is odd cases , then adj (A) = ..

Step-by-step solution

Given A = [a_ ij ]_ 3 3 where a_ ij = 1 if i+j is even, and 0 if i+j is odd. Constructing the matrix A : A = bmatrix 1 & 0 & 1 0 & 1 & 0 1 & 0 & 1 bmatrix The adjoint of A is the transpose of the cofactor matrix C . Calculating the cofactors C_ ij = (-1)^ i+j M_ ij : C₁₁ = 1, C₁₂ = 0, C₁₃ = -1 C₂₁ = 0, C₂₂ = 0, C₂₃ = 0 C₃₁ = -1, C₃₂ = 0, C₃₃ = 1 The cofactor matrix is C = bmatrix 1 & 0 & -1 0 & 0 & 0 -1 & 0 & 1 bmatrix . Taking the transpose of C gives the adjoint matrix: adj (A) = C^T = bmatrix 1 & 0 & -1 0 & 0 &