If A = 1/9 [(-8 1 4), (1 3 1), (-1 1 3)], prove that A^–1 = A^T .

Question

If \(A=\frac{1}{9}\begin{bmatrix}-8&1&4\\4&4&7\\1&-8&4\end{bmatrix}\), 

prove that A^–1 = A^T .

Answer 1

\(A=\frac{1}{9}\begin{bmatrix}-8&1&4\\4&4&7\\1&-8&4\end{bmatrix}\) A^T = \(\frac{1}{9}\begin{bmatrix}-8&1&4\\4&4&7\\1&-8&4\end{bmatrix}\)

|A| = \(\frac{1}{9}\)[ – 8(16 + 56) – 1(16 – 7) + 4( – 32 – 4)]

= – 81

Cofactors of A are:

C₁₁ = 72 C₂₁ = – 36 C₃₁ = – 9

C₁₂ = – 9 C₂₂ = – 36 C₃₂ = 72

C₁₃ = – 36 C₂₃ = – 63 C₃₃ = – 36