(a)

\begin{align*} \left( \begin{matrix} 3 & 2 \\ 4 & - 1 \end{matrix} \right) + \left( \begin{matrix} 5 & 4 \\ 3 & 2 \end{matrix} \right) - \left( \begin{matrix} 6 & 3 \\ 1 & - 2 \end{matrix} \right) & = \left( \begin{matrix} 8 & 6 \\ 7 & 1 \end{matrix} \right) - \left( \begin{matrix} 6 & 3 \\ 1 & -2 \end{matrix} \right) \\ & = \left( \begin{matrix} 2 & 3 \\ 6 & 3 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \left( \begin{matrix} 2 & 3 & -4 \\ 6 & -1 & 3 \end{matrix} \right) + \left( \begin{matrix} 3 & 1 & 5 \\ -3 & 2 & 7 \end{matrix} \right) + \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \end{matrix} \right) & = \left( \begin{matrix} 5 & 4 & 1 \\ 3 & 1 & 10 \end{matrix} \right) + \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 6 & 4 & 1 \\ 3 & 2 & 11 \end{matrix} \right) \end{align*}

(c)

\begin{align*} \left( \begin{matrix} 2 & 3 \\ 4 & -7 \\ 5 & -3 \end{matrix} \right) - \left( \begin{matrix} 4 & 5 \\ -2 & 7 \\ 6 & -1 \end{matrix} \right) + \left( \begin{matrix} -3 & 4 \\ -1 & 7 \\ -6 & 2 \end{matrix} \right) & = \left( \begin{matrix} -2 & -2 \\ 6 & -14 \\ -1 & -2 \end{matrix} \right) + \left( \begin{matrix} -3 & 4 \\ -1 & 7 \\ -6 & 2 \end{matrix} \right) \\ & = \left( \begin{matrix} -5 & 2 \\ 5 & -7 \\ -7 & 0 \end{matrix} \right) \end{align*}

(d)

\begin{align*} \left( \begin{matrix} 3 \\ 5 \end{matrix} \right) - \left( \begin{matrix} 4 \\ 7 \end{matrix} \right) + \left( \begin{matrix} 5 \\ -3 \end{matrix} \right) & = \left( \begin{matrix} -1 \\ -2 \end{matrix} \right) + \left( \begin{matrix} 5 \\ -3 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 \\ -5 \end{matrix} \right) \end{align*}

(e)

\begin{align*} \left( \begin{matrix} 1 & 3 \end{matrix} \right) - \left( \begin{matrix} 3 & 2 \end{matrix} \right) + \left( \begin{matrix} 6 & 5 \end{matrix} \right) & = \left( \begin{matrix} -2 & 1 \end{matrix} \right) + \left( \begin{matrix} 6 & 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 & 6 \end{matrix} \right) \end{align*}

(f)

\begin{align*} \left( \begin{matrix} 1 & 0 & 7 \end{matrix} \right) + \left( \begin{matrix} 3 & -2 & 4 \end{matrix} \right) - \left( \begin{matrix} 7 & 3 & -5 \end{matrix} \right) & = \left( \begin{matrix} 4 & -2 & 11 \end{matrix} \right) - \left( \begin{matrix} 7 & 3 & -5 \end{matrix} \right) \\ & = \left( \begin{matrix} -3 & -5 & 16 \end{matrix} \right) \end{align*}

(i)

\begin{align*} 2\textbf{A} + \textbf{B} & = \textbf{C} \\ 2 \left( \begin{matrix} 2 & -1 \\ 1 & 3 \end{matrix} \right) + \left( \begin{matrix} 5 & a \\ c & 4 \end{matrix} \right) & = \left( \begin{matrix} b & 6 \\ 4 & d \end{matrix} \right) \\ \left( \begin{matrix} 4 & -2 \\ 2 & 6 \end{matrix} \right) + \left( \begin{matrix} 5 & a \\ c & 4 \end{matrix} \right) & = \left( \begin{matrix} b & 6 \\ 4 & d \end{matrix} \right) \\ \left( \begin{matrix} 9 & -2 + a \\ 2 + c & 10 \end{matrix} \right) & = \left( \begin{matrix} b & 6 \\ 4 & d \end{matrix} \right) \\ \\ b & = 9 \\ \\ -2 + a & = 6 \\ a & = 6 + 2 \\ a & = 8 \\ \\ 2 + c & = 4 \\ c & = 4 - 2 \\ c & = 2 \\ \\ 10 & = d \end{align*}

(ii)

\begin{align*} 3\textbf{A} - 2\textbf{B} & = 4\textbf{C} \\ \\ 3\left( \begin{matrix} 2 & -1 \\ 1 & 3 \end{matrix} \right) - 2\left( \begin{matrix} 5 & a \\ c & 4 \end{matrix} \right) & = 4 \left( \begin{matrix} b & 6 \\ 4 & d \end{matrix} \right) \\ \left( \begin{matrix} 6 & -3 \\ 3 & 9 \end{matrix} \right) - \left( \begin{matrix} 10 & 2a \\ 2c & 8 \end{matrix} \right) & = \left( \begin{matrix} 4b & 24 \\ 16 & 4d \end{matrix} \right) \\ \left( \begin{matrix} -4 & -3 -2a \\ 3 -2c & 1 \end{matrix} \right) & = \left( \begin{matrix} 4b & 24 \\ 16 & 4d \end{matrix} \right) \\ \\ -4 & = 4b \\ {-4 \over 4} & = b \\ -1 & = b \\ \\ -3 - 2a & =24 \\ -2a & =24 + 3 \\ -2a & = 27 \\ a & = {27 \over -2} \\ a & = -13{1 \over 2} \\ \\ 3 - 2c & = 16 \\ -2c & = 16 - 3 \\ -2c & = 13 \\ c & = {13 \over -2} \\ c & = -6{1 \over 2} \\ \\ 1 & = 4d \\ {1 \over 4} & = d \end{align*}

(a)

\begin{align*} \underset{2 \times 1}{\left( \begin{matrix} 1 \\ 3 \end{matrix} \right)} \underset{1 \times 2}{\left( \begin{matrix} 3 & 1 \end{matrix} \right)} & = \left( \begin{matrix} 1 \times 3 & 1 \times 1 \\ 3 \times 3 & 3 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 3 & 1 \\ 9 & 3 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \underset{1 \times 2}{\left( \begin{matrix} 2 & 3 \end{matrix} \right)} \underset{2 \times 1}{\left( \begin{matrix} 3 \\ 1 \end{matrix} \right)} & = \left( \begin{matrix} 2 \times 3 + 3 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 9 \end{matrix} \right) \end{align*}

(c)

\begin{align*} \underset{1 \times 3}{\left( \begin{matrix} 1 & 2 & 3 \end{matrix} \right)} \underset{3 \times 1}{\left( \begin{matrix} 3 \\ 2 \\ 1 \end{matrix} \right)} & = \left( \begin{matrix} 1 \times 3 + 2 \times 2 + 3 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 10 \end{matrix} \right) \end{align*}

(d)

\begin{align*} \underset{3 \times 1}{\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right)} \underset{1 \times 3}{\left( \begin{matrix} 3 & 0 & 1 \end{matrix} \right)} & = \left( \begin{matrix} 1 \times 3 & 1 \times 0 & 1 \times 1 \\ 2 \times 3 & 2 \times 0 & 2 \times 1 \\ 3 \times 3 & 3 \times 0 & 3 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 3 & 0 & 1 \\ 6 & 0 & 2 \\ 9 & 0 & 3 \end{matrix} \right) \end{align*}

(e)

\begin{align*} & \underset{2 \times 2}{\left( \begin{matrix} 7 & 9 \\ 3 & -5 \end{matrix} \right)} \underset{1 \times 2}{\left( \begin{matrix} 2 & 6 \end{matrix} \right)} \\ \\ & \text{Not possible} \end{align*}

(f)

\begin{align*} \underset{2 \times 2}{\left( \begin{matrix} -2 & 3 \\ -1 & -2 \end{matrix} \right)} \underset{2 \times 1}{\left( \begin{matrix} {1 \over 2} \\ 1{1 \over 2} \end{matrix} \right)} & = \left( \begin{matrix} (-2) \times {1 \over 2} + 3 \times 1{1 \over 2} \\ (-1) \times {1 \over 2} + (-2) \times 1{1 \over 2} \end{matrix} \right) \\ & = \left( \begin{matrix} 3{1 \over 2} \\ -3{1 \over 2} \end{matrix} \right) \end{align*}

(g)

\begin{align*} \underset{3 \times 2}{\left( \begin{matrix} 0 & 2 \\ 3 & 1 \\ -1 & 1 \end{matrix} \right)} \underset{2 \times 1}{\left( \begin{matrix} 2 \\ 1 \end{matrix} \right)} & = \left( \begin{matrix} 0 \times 2 + 2 \times 1 \\ 3 \times 2 + 1 \times 1 \\ (-1) \times 2 + 1 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 2 \\ 7 \\ -1 \end{matrix} \right) \end{align*}

(h)

\begin{align*} \underset{3 \times 2}{\left( \begin{matrix} 0 & -2 \\ -1 & 1 \\ 3 & -1 \end{matrix} \right)} \underset{2 \times 2}{\left( \begin{matrix} 2 & 1 \\ 0 & -4 \end{matrix} \right)} & = \left( \begin{matrix} 0 \times 2 + (-2) \times 0 & 0 \times 1 + (-2) \times (-4) \\ (-1) \times 2 + 1 \times 0 & (-1) \times 1 + 1 \times (-4) \\ 3 \times 2 + (-1) \times 0 & 3 \times 1 + (-1) \times (-4) \end{matrix} \right) \\ & = \left( \begin{matrix} 0 & 8 \\ -2 & -5 \\ 6 & 7 \end{matrix} \right) \end{align*}

(i)

\begin{align*} \underset{2 \times 3}{\left( \begin{matrix} 2 & 1 & 3 \\ -1 & -1 & 4 \end{matrix} \right)} \underset{3 \times 1}{\left( \begin{matrix} 1 \\ 2 \\ -1 \end{matrix} \right)} & = \left( \begin{matrix} 2 \times 1 + 1 \times 2 + 3 \times (-1) \\ (-1) \times 1 + (-1) \times 2 + 4 \times (-1) \end{matrix} \right) \\ & = \left( \begin{matrix} 1 \\ -7 \end{matrix} \right) \end{align*}

(j)

\begin{align*} \underset{1 \times 3}{\left( \begin{matrix} 1 & 3 & 2 \end{matrix} \right)} \underset{3 \times 2}{\left( \begin{matrix} 3 & -2 \\ 1 & 4 \\ -1 & 2 \end{matrix} \right)} & = \left( \begin{matrix} 1 \times 3 + 3 \times 1 + 2 \times (-1) & 1 \times (-2) + 3 \times 4 + 2 \times 2 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 & 14 \end{matrix} \right) \end{align*}

(a)

\begin{align*} \left( \begin{matrix} 1 & 3 & 2 \\ 0 & 1 & -2 \end{matrix} \right) \left( \begin{matrix} 1 \\ a \\ 2 \end{matrix} \right) & = \left( \begin{matrix} 5 \\ b \end{matrix} \right) \\ \left( \begin{matrix} 1 \times 1 + 3 \times a + 2 \times 2 \\ 0 \times 1 + 1 \times a + (-2) \times 2 \end{matrix} \right) & = \left( \begin{matrix} 5 \\ b \end{matrix} \right) \\ \left( \begin{matrix} 5 + 3a \\ a - 4 \end{matrix} \right) & = \left( \begin{matrix} 5 \\ b \end{matrix} \right) \\ \\ 5 + 3a & = 5 \\ 3a & = 5 - 5 \\ 3a & = 0 \\ a & = {0 \over 3} \\ a & = 0 \\ \\ a - 4 & = b \\ (0) - 4 & = b \\ -4 & = b \end{align*}

(b)

\begin{align*} \left( \begin{matrix} x \\ y \end{matrix} \right) & = \left( \begin{matrix} 2 & 1 \\ 3 & 0 \end{matrix} \right) \left( \begin{matrix} 2 \\ 5 \end{matrix} \right) + \left( \begin{matrix} 3 \\ 2\end{matrix} \right) \\ \left( \begin{matrix} x \\ y \end{matrix} \right) & = \left( \begin{matrix} 2 \times 2 + 1 \times 5 \\ 3 \times 2 + 0 \times 5 \end{matrix} \right) + \left( \begin{matrix} 3 \\ 2 \end{matrix} \right) \\ \left( \begin{matrix} x \\ y \end{matrix} \right) & = \left( \begin{matrix} 9 \\ 6 \end{matrix} \right) + \left( \begin{matrix} 3 \\ 2 \end{matrix} \right) \\ \left( \begin{matrix} x \\ y \end{matrix} \right) & = \left( \begin{matrix} 12 \\ 8 \end{matrix} \right) \\ \\ x & = 12, y = 8 \end{align*}

(c)

\begin{align*} \left( \begin{matrix} 0 & 1 \\ -2 & 0 \end{matrix} \right) \left( \begin{matrix} a & -4 \\ b & 0 \end{matrix} \right) & = \left( \begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix} \right) + \left( \begin{matrix} 0 & -3 \\ 6 & 2c \end{matrix} \right) \\ \left( \begin{matrix} 0 \times a + 1 \times b & 0 \times (-4) + 1 \times 0 \\ (-2) \times a + 0 \times b & (-2) \times (-4) + 0 \times 0 \end{matrix} \right) & = \left( \begin{matrix} 2 & 0 \\ 6 & 1 + 2c \end{matrix} \right) \\ \left( \begin{matrix} b & 0 \\ -2a & 8 \end{matrix} \right) & = \left( \begin{matrix} 2 & 0 \\ 6 & 1 + 2c \end{matrix} \right) \\ \\ b & = 2 \\ \\ -2a & = 6 \\ a & = {6 \over -2} \\ a & = -3 \\ \\ 8 & = 1 + 2c \\ 8 - 1 & = 2c \\ 7 & = 2c \\ {7 \over 2} & = c \end{align*}

\begin{align*} \left( \begin{matrix} 450 & 240 & 120 & 80 & 60 \\ 250 & 140 & 80 & 60 & 20 \\ 280 & 120 & 50 & 30 & 24 \end{matrix} \right) \left( \begin{matrix} 1 \\ 1.5 \\ 6.5 \\ 5.5 \\ 4.8 \end{matrix} \right) & = \left( \begin{matrix} 450 \times 1 + 240 \times 1.5 + 120 \times 6.5 + 80 \times 5.5 + 60 \times 4.8 \\ 250 \times 1 + 140 \times 1.5 + 80 \times 6.5 + 60 \times 5.5 + 20 \times 4.8 \\ 280 \times 1 + 120 \times 1.5 + 50 \times 6.5 + 30 \times 5.5 + 24 \times 4.8 \end{matrix} \right) \\ & = \left( \begin{matrix} 2318 \\ 1406 \\ 1065.2 \end{matrix} \right) \end{align*}

(i) Note Shop B did not order any Sprite and Shop C did not order any Coke

\begin{align*} \left( \begin{matrix} 12 & 8 & 12 & 15 \\ 15 & 0 & 16 & 14 \\ 0 & 20 & 25 & 16 \end{matrix} \right) \left( \begin{matrix} 8.4 \\ 7.8 \\ 8.8 \\ 8.2 \end{matrix} \right) & = \left( \begin{matrix} 12 \times 8.4 + 8 \times 7.8 + 12 \times 8.8 + 15 \times 8.2 \\ 15 \times 8.4 + 0 + 16 \times 8.8 + 14 \times 8.2 \\ 0 + 20 \times 7.8 + 25 \times 8.8 + 16 \times 8.2 \end{matrix} \right) \\ & = \left( \begin{matrix} 391.8 \\ 381.6 \\ 507.2 \end{matrix} \right) \end{align*}

(ii)

\begin{align} \left( \begin{matrix} 22 & 18 & 25 \end{matrix} \right) \left( \begin{matrix} 391.8 \\ 381.6 \\ 507.2 \end{matrix} \right) & = \left( \begin{matrix} 391.8 \times 22 + 381.6 \times 18 + 507.2 \times 25 \end{matrix} \right) \\ & = \left( \begin{matrix} 28 \phantom{.} 168.4 \end{matrix} \right) \\ \\ \text{Total amount collected} & = \$ 28 \phantom{.} 168.40 \end{align}

\begin{align*} \left( \begin{matrix} 480 & 460 & 620 & 430 \\ 350 & 450 & 385 & 540 \\ 420 & 520 & 420 & 620 \\ 380 & 452 & 250 & 486 \end{matrix} \right) \left( \begin{matrix} 0.10 \\ 0.20 \\ 0.50 \\ 1 \end{matrix} \right) & = \left( \begin{matrix} 480 \times 0.10 + 460 \times 0.20 + 620 \times 0.50 + 430 \times 1 \\ 350 \times 0.10 + 450 \times 0.20 + 385 \times 0.50 + 540 \times 1 \\ 420 \times 0.10 + 520 \times 0.20 + 420 \times 0.50 + 620 \times 1 \\ 380 \times 0.10 + 452 \times 0.20 + 250 \times 0.50 + 486 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 880 \\ 857.5 \\ 976 \\ 739.4 \end{matrix} \right) \\ \\ \left( \begin{matrix} 1 & 1 & 1 & 1 \end{matrix} \right) \left( \begin{matrix} 880 \\ 857.5 \\ 976 \\ 739.4 \end{matrix} \right) & = \left( \begin{matrix} 1 \times 880 + 1 \times 857.5 + 1 \times 976 + 1 \times 739.4 \end{matrix} \right) \\ & = \left( \begin{matrix} 3 \phantom{.} 452.9 \end{matrix} \right) \\ \\ \text{Total amount} & = \$ 3 \phantom{.} 452.90 \end{align*}

(i)

\begin{align*} \left( \begin{matrix} 11 & 2 & 5 \\ 7 & 2 & 11 \\ 4 & 5 & 10 \\ 7 & 4 & 7 \\ 12 & 1 & 9 \\ 9 & 2 & 8 \end{matrix} \right) \left( \begin{matrix} 3 \\ 1 \\ 0 \end{matrix} \right) & = \left( \begin{matrix} 11 \times 3 + 2 \times 1 + 0 \\ 7 \times 3 + 2 \times 1 + 0 \\ 4 \times 3 + 5 \times 1 + 0 \\ 7 \times 3 + 4 \times 1 = 0 \\ 12 \times 3 + 1 \times 1 + 0 \\ 9 \times 3 + 2 \times 1 + 0 \end{matrix} \right) \\ & = \left( \begin{matrix} 35 \\ 23 \\ 17 \\ 25 \\ 37 \\ 29 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \left( \begin{matrix} 18 & 11 & 2 & 5 \\ 20 & 7 & 2 & 11 \\ 19 & 4 & 5 & 10 \\ 18 & 7 & 4 & 7 \\ 22 & 12 & 1 & 9 \\ 19 & 9 & 2 & 8 \end{matrix} \right) \left( \begin{matrix} 300 \\ 500 \\ 200 \\ -300 \end{matrix} \right) & = \left( \begin{matrix} 18 \times 300 + 11 \times 500 + 2 \times 200 + 5 \times (-300) \\ 20 \times 300 + 7 \times 500 + 2 \times 200 + 11 \times (-300) \\ 19 \times 300 + 4 \times 500 + 5 \times 200 + 10 \times (-300) \\ 18 \times 300 + 7 \times 500 + 4 \times 200 + 7 \times (-300) \\ 22 \times 300 + 12 \times 500 + 1 \times 200 + 9 \times (-300) \\ 19 \times 300 + 9 \times 500 + 2 \times 200 + 8 \times (-300) \end{matrix} \right) \\ & = \left( \begin{matrix} 9 \phantom{.} 800 \\ 6 \phantom{.} 600 \\ 5 \phantom{.} 700 \\ 7 \phantom{.} 600 \\ 10 \phantom{.} 100 \\ 8 \phantom{.} 200 \end{matrix} \right) \end{align*}