Given the matrices :

$\text { A } = \begin{bmatrix$}[r] 2 & 1 \ 4 & 2 \end{bmatrix}, \text{ B } = \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \text{ and C } = \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} .

Find the products of (i) ABC (ii) ACB and state whether they are equal.

(i)

$\text{ABC } = \begin{bmatrix} 2 & 1 \ 4 & 2 \end{bmatrix}\begin{bmatrix$}[r] 3 & 4 \ -1 & -2 \end{bmatrix}\begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 \times 3 + 1 \times (-1) & 2 \times 4 + 1 \times (-2) \ 4 \times 3 + 2 \times (-1) & 4 \times 4 + 2 \times (-2) \end{bmatrix} \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 6 - 1 & 8 - 2 \ 12 - 2 & 16 - 4 \end{bmatrix} \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix} 5 & 6 \ 10 & 12 \end{bmatrix} \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 5 \times (-3) + 6 \times 0 & 5 \times 1 + 6 \times (-2) \ 10 \times (-3) + 12 \times 0 & 10 \times 1 + 12 \times (-2) \end{bmatrix} \\[1em] = \begin{bmatrix} -15 + 0 & 5 - 12 \ -30 + 0 & 10 + (-24) \end{bmatrix} \\[1em] = \begin{bmatrix} -15 & -7 \ -30 & -14 \end{bmatrix} \\[1em]

Hence, the matrix ABC = $\begin{bmatrix} -15 & -7 \ -30 & -14 \end{bmatrix} .$

(ii)

$\text{ACB } = \begin{bmatrix$}[r] 2 & 1 \ 4 & 2 \end{bmatrix} \begin{bmatrix}[r] -3 & 1 \ 0 & -2 \end{bmatrix} \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 \times (-3) + 1 \times 0 & 2 \times 1 + 1 \times (-2) \ 4 \times (-3) + 2 \times 0 & 4 \times 1 + 2 \times (-2) \end{bmatrix} \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -6 + 0 & 2 - 2 \ -12 + 0 & 4 - 4 \end{bmatrix} \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -6 & 0 \ -12 & 0 \end{bmatrix} \begin{bmatrix}[r] 3 & 4 \ -1 & -2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -6 \times 3 + 0 \times (-1) & -6 \times 4 + 0 \times (-2) \ -12 \times 3 + 0 \times (-1) & -12 \times 4 + 0 \times (-2) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -18 + 0 & -24 + 0 \ -36 + 0 & -48 + 0 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -18 & -24 \ -36 & -48 \end{bmatrix} \\[1em]

Hence, the matrix ACB = $\begin{bmatrix} -18 & -24 \ -36 & -48 \end{bmatrix} ,$ and matrix ABC ≠ ACB.