91Ó°ÊÓ

The inverse of an\(m \times m\)matrix A can be computed by using the augmented matrix\(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\), where\(I\)is the identity matrix. Matrix Ahas an inverse only if \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\) is row equivalent to \(\left( {\begin{aligned}{*{20}{c}}I&{{A^{ - 1}}}\end{aligned}} \right)\).

Product AB is invertible, so there should be an inverse of matrix AB. Let D be the inverse matrix of AB.

Then, it can be represented as shown below:

\(\begin{aligned}{c}D\left( {AB} \right) = I\\\left( {DA} \right)B = I\end{aligned}\)

The equation\(\left( {DA} \right)B = I\) shows that matrix\(DA\)is the inverse of matrix B.

Therefore, B is invertible.