Find the inverse of the matrix [(a b), (c 1+bc/a)] and show that aA^– 1 = (a^2 + bc + 1) I – aA.

Question

Find the inverse of the matrix \(A=\begin{bmatrix}a&b\\c&\frac{1+bc}{a}\end{bmatrix}\) and show that aA ^–1 = (a² + bc + 1) I – aA.

Answer 1

\(A=\begin{bmatrix}a&b\\c&\frac{1+bc}{a}\end{bmatrix}\)

Now, |A| = bc =\(\frac{a+abc}{a}-bc\)\(=\frac{a+abc-abc}{a}\)\(=1\neq0\)

Hence, A^–1 exists.

Cofactors of A are

C₁₁ = \(\frac{1+bc}{a}\) C₁₂ = – c

C₂₁ = – b C₂₂ = a

Hence, LHS = RHS