Verify A (adj A) = (adj A) A = |A|

Adjoint of the matrix A = [a_ij]_n×n is defined as the transpose of the matrix [A_ij]_n×n where A_ij is the co-factor of the element a_ij.

Let’s find the cofactors for all the positions first-

Here, A₁₁ = 0, A₁₂ = -11, A₁₃ = 0, A₂₁ = 3, A₂₂ = 1, A₂₃ = -1, A₃₁ = 2, A₃₂ = 8, A₃₃ = 3.

∴ Adj A =

=

So, LHS = A(AdjA) =

Also AdjA(A) =

Determinant of A = |A| = 11

So RHS = |A|I = .

Hence A(AdjA) = AdjA(A) = |A|I = {hence proved}