Find the adjoint of the matrix A = [(-1, -2, -2), (2, 1, -2), (2, -2 ,1)]and hence show that A(adj A) = |A| I3.

Question

Find the adjoint of the matrix \(A=\begin{bmatrix}-1&-2&-2\\2&1&-2\\2&-2&1\end{bmatrix}\) and hence show that A(adj A) = |A| I₃.

Answer 1

= – 1(1 – 4) + 2(2 + 4) – 2( – 4 – 2)

= 3 + 12 + 12

|A| = 27

Cofactors of A

C₁₁ = – 3 C₂₁ = – 6 C₃₁ = 6

C₁₂ = – 6 C₂₂ = 3 C₃₂ = – 6

C₁₃ = – 6 C₂₃ = – 6 C₃₃ = 3

Hence, A(adj A) = |A|I