For the following pairs of matrices verify that (AB)^–1 = B^–1A^–1: A = [(2 1),(5 3)] and B = [(4 5), (3 4)]

Question

For the following pairs of matrices verify that (AB)^–1 = B ^–1A ^–1:

\(A=\begin{bmatrix}2&1\\5&3\end{bmatrix}\) and \(B=\begin{bmatrix}4&5\\3&4\end{bmatrix}\)

Answer 1

|A| = 1

Adj A =\(\begin{bmatrix}3&-1\\-5&2\end{bmatrix}\)

A ^–1 = \(\frac{adjA}{|A|}\)\(=\frac{1}{1}\begin{bmatrix}3&-1\\-5&2\end{bmatrix}\)

\(B=\begin{bmatrix}4&5\\3&4\end{bmatrix}\) 

|B| = – 1

Hence, (AB) ^–1 = B ^–1A ^–1.