Solution:

Two matrices A and B are the multiplicative inverse of each other if their product is the identity matrix.

AB = $I$ = $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$

 

(i) $\left[ \begin{matrix} 3 & 5 \\ 4 & 7 \\ \end{matrix} \right]$ and $\left[ \begin{matrix} 7 & -5 \\ -4 & 3 \\ \end{matrix} \right]$

Let A = $\left[ \begin{matrix} 3 & 5 \\ 4 & 7 \\ \end{matrix} \right]$ and B = $\left[ \begin{matrix} 7 & -5 \\ -4 & 3 \\ \end{matrix} \right]$

 

AB = $\left[ \begin{matrix} 3 & 5 \\ 4 & 7 \\ \end{matrix} \right]\left[ \begin{matrix} 7 & -5 \\ -4 & 3 \\ \end{matrix} \right]$

= $\left[ \begin{matrix} 3\times 7+5~\times \left( -4 \right) & 3\times \left( -5 \right)+5~\times 3 \\ 4\times 7+7~\times \left( -4 \right) & 4\times \left( -5 \right)+7~\times 3 \\ \end{matrix} \right]$

= $\left[ \begin{matrix} 21-20 & -15+15 \\ 28-28 & -20+21 \\ \end{matrix} \right]$

= $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$ = $I$

 

Hence the given matrices are multiplicative inverses of each other.

 

(ii) $\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \\ \end{matrix} \right]$ and $\left[ \begin{matrix} -3 & 2 \\ 2 & -1 \\ \end{matrix} \right]$

Let A = $\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \\ \end{matrix} \right]$ and B = $\left[ \begin{matrix} -3 & 2 \\ 2 & -1 \\ \end{matrix} \right]$

 

AB = $\left[ \begin{matrix} 1 & 2 \\ 2 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} -3 & 2 \\ 2 & -1 \\ \end{matrix} \right]$

= $\left[ \begin{matrix} 1\times \left( -3 \right)+2~\times 2 & 1\times 2+2~\times \left( -1 \right) \\ 2\times \left( -3 \right)+3~\times 2 & 2\times 2+3\times \left( -1 \right) \\ \end{matrix} \right]$

= $\left[ \begin{matrix} -3+4 & 2-2 \\ -6+6 & 4-3 \\ \end{matrix} \right]$

= $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$ = $I$

 

Hence the given matrices are multiplicative inverses of each other.