Question

For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]

\[A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}\]

Answer 1

\[\text{ We have, }A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B = \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}\]
\[ \therefore AB = \begin{bmatrix}11 & 14 \\ 29 & 37\end{bmatrix}\]
Now,
\[\left| AB \right| = 1\]
\[\text{ Since, }\left| AB \right| \neq 0\]
\[\text{ Hence, AB is invertible . Let }C_{ij}\text{ be the cofactor of }a_{in}\text{ in AB = }\left[ a_{ij} \right]\]
\[ C_{11} = 37 , C_{12} = - 29, C_{21} = - 14\text{ and }C_{22} = 11\]
\[adj(AB) = \begin{bmatrix}37 & - 14 \\ - 29 & 11\end{bmatrix}\]
\[ \therefore \left( AB \right)^{- 1} = \begin{bmatrix}37 & - 14 \\ - 29 & 11\end{bmatrix} . . . \left( 1 \right)\]
\[\left| B \right| = 1\]
\[\text{  Since, }\left| B \right| \neq 0\]
\[\text{ Hence, B is invertible . Let }C_{ij}\text{ be the cofactor of }a_{in}\text{ in B = }\left[ a_{ij} \right]\]
\[ C_{11} = 4 , C_{12} = - 3, C_{21} = - 5\text{ and }C_{22} = 4\]
\[adjB = \begin{bmatrix}4 & - 5 \\ - 3 & 4\end{bmatrix}\]
\[ \therefore B^{- 1} = \begin{bmatrix}4 & - 5 \\ - 3 & 4\end{bmatrix}\]
\[\left| A \right| = 1\]
\[\text{ Since, }\left| A \right| \neq 0\]
\[\text{ Hence, A is invertible . Let }C_{ij}\text{ be the cofactor of }a_{in}\text{ in A = }\left[ a_{ij} \right]\]
\[ C_{11} = 3 , C_{12} = - 5, C_{21} = - 1\text{ and }C_{22} = 2\]
\[adjA = \begin{bmatrix}3 & - 1 \\ - 5 & 2\end{bmatrix}\]
\[ \therefore A^{- 1} = \begin{bmatrix}3 & - 1 \\ - 5 & 2\end{bmatrix}\]
\[\text{ Now, }B^{- 1} A^{- 1} = \begin{bmatrix}37 & - 14 \\ - 29 & 11\end{bmatrix} . . . \left( 2 \right)\]
\[\text{ From eq . }\left( 1 \right)\text{ and }\left( 2 \right),\text{ we have}\]
\[ \left( AB \right)^{- 1} = B^{- 1} A^{- 1} \]
Hence verified .