9 Matrices \(\mathbf { M }\) and \(\mathbf { N }\) are given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 2
0 & 1 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { r r } 1 & - 3
1 & 4 \end{array} \right)\).
- Find \(\mathbf { M } ^ { - 1 }\) and \(\mathbf { N } ^ { - 1 }\).
- Find \(\mathbf { M N }\) and \(( \mathbf { M N } ) ^ { - \mathbf { 1 } }\). Verify that \(( \mathbf { M N } ) ^ { - 1 } = \mathbf { N } ^ { - 1 } \mathbf { M } ^ { - 1 }\).
- The result \(( \mathbf { P Q } ) ^ { - 1 } = \mathbf { Q } ^ { - 1 } \mathbf { P } ^ { - 1 }\) is true for any two \(2 \times 2\), non-singular matrices \(\mathbf { P }\) and \(\mathbf { Q }\).
The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof.
$$\begin{aligned}
& ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q } = \mathbf { I }
\Rightarrow & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q Q } \mathbf { Q } ^ { - 1 } = \mathbf { I Q } ^ { - 1 }
\end{aligned}$$