If A = [(0, 1, 1), (1, 0, 1), (1, 1, 0)], find A^–1 and show that A^–1 = 1/2(A^2 – 3I)

Question

If \(A=\begin{bmatrix}0&1&1\\1&0&1\\1&1&0\end{bmatrix}\), find A^–1 and show that A^–1 = 1/2(A² – 3I).

Answer 1

\(A=\begin{bmatrix}0&1&1\\1&0&1\\1&1&0\end{bmatrix}\)|A| = 0 – 1(0 – 1) + 1(1 – 0) = 0 + 1 + 1 = 2

Cofactors of A are:

C₁₁ = – 1 C₂₁ = 1 C₃₁ = 1

C₁₂ = 1 C₂₂ = – 1 C₃₂ = 1

C₁₃ = 1 C₂₃ = 1 C₃₃ = – 1

Hence, A^–1 = (A² – 3I)