(a)

\begin{align*} \left( \begin{matrix} -1 & 1 \\ -2 & 5 \end{matrix} \right) + \left( \begin{matrix} 6 & 0 \\ 1 & 4 \end{matrix} \right) & = \left( \begin{matrix} 5 & 1 \\ -1 & 9 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \underset{3 \times 1}{ \left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) } - \underset{3 \times 1}{ \left( \begin{matrix} 3 \\ 2 \\ 1 \end{matrix} \right) } & + \underset{2 \times 1}{ \left( \begin{matrix} -1 \\ -2 \end{matrix} \right) } \\ \\ \text{Not possible since } & \text{matrices have different order} \end{align*}

(c)

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

(d)

\begin{align*} (3) + {1 \over 2}(2) - (0) & = (3) + (1) - (0) \\ & = (4) \end{align*}

(a)

\begin{align*} \underset{2 \times 1}{ \left( \begin{matrix} 5 \\ -1 \end{matrix} \right) } \underset{1 \times 2}{ \left( \begin{matrix} 3 & 4 \end{matrix} \right) } & = \underset{2 \times 2}{ \left( \begin{matrix} 5 \times 3 & 5 \times 4 \\ (-1) \times 3 & (-1) \times 4 \end{matrix} \right) } \\ & = \left( \begin{matrix} 15 & 20 \\ -3 & -4 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \underset{1 \times 2}{ \left( \begin{matrix} 3 & 2 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} 4 \\ -1 \end{matrix} \right) } & = \underset{1 \times 1}{ \left( \begin{matrix} (3 \times 4) + [2 \times (-1)] \end{matrix} \right) } \\ & = \left( \begin{matrix} 10 \end{matrix} \right) \end{align*}

(c)

\begin{align*} \underset{3 \times 2}{ \left( \begin{matrix} 3 & 4 \\ 1 & 5 \\ 2 & 6 \end{matrix} \right) } & \underset{3 \times 1}{ \left( \begin{matrix} 3 \\ 1 \\ 5 \end{matrix} \right) } \\ \\ \text{Not possible since the columns of the first matrix} & \text{ is not equals to the rows of the second matrix} \end{align*}

(d)

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

(e)

\begin{align*} \underset{2 \times 2}{ \left( \begin{matrix} 4 & 1 \\ 0 & 2 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) } & = \underset{2 \times 2}{ \left( \begin{matrix} (4 \times 1) + (1 \times 3) & (4 \times 2) + (1 \times 4) \\ (0 \times 1) + (2 \times 3) & (0 \times 2) + (2 \times 4) \end{matrix} \right) } \\ & = \left( \begin{matrix} 7 & 12 \\ 6 & 8 \end{matrix} \right) \end{align*}

(i)

\begin{align*} 2 \left( \begin{matrix} 2 & -1 \\ 1 & 3 \end{matrix} \right) + \left( \begin{matrix} 5 & w \\ y & 4 \end{matrix} \right) & = \left( \begin{matrix} x & 6 \\ 4 & z \end{matrix} \right) \\ \left( \begin{matrix} 4 & -2 \\ 2 & 6 \end{matrix} \right) + \left( \begin{matrix} 5 & w \\ y & 4 \end{matrix} \right) & = \left( \begin{matrix} x & 6 \\ 4 & z \end{matrix} \right) \\ \left( \begin{matrix} 9 & - 2 + w \\ 2 + y & 10 \end{matrix} \right) & = \left( \begin{matrix} x & 6 \\ 4 & z \end{matrix} \right) \end{align*} \begin{align*} x & = 9 &&& -2 + w & = 6 &&& 2 + y & = 4 &&& z & = 10 \\ & &&& w & = 8 &&& y & = 2 &&& \end{align*}

(ii)

\begin{align*} 3 \left( \begin{matrix} 2 & -1 \\ 1 & 3 \end{matrix} \right) - 2 \left( \begin{matrix} 5 & w \\ y & 4 \end{matrix} \right) & = 4 \left( \begin{matrix} x & 6 \\ 4 & z \end{matrix} \right) \\ \left( \begin{matrix} 6 & - 3 \\ 3 & 9 \end{matrix} \right) - \left( \begin{matrix} 10 & 2w \\ 2y & 8 \end{matrix} \right) & = \left( \begin{matrix} 4x & 24 \\ 16 & 4z \end{matrix} \right) \\ \left( \begin{matrix} -4 & -3 - 2w \\ 3 - 2y & 1 \end{matrix} \right) & = \left( \begin{matrix} 4x & 24 \\ 16 & 4z \end{matrix} \right) \end{align*} \begin{align*} -4 & = 4x &&& -3 - 2w & = 24 &&& 3 - 2y & = 16 &&& 1 & = 4z \\ -1 & = x &&& -2w & = 27 &&& -2y & = 13 &&& {1 \over 4} & = z \\ & &&& w & = -{27 \over 2} &&& y & = -{13 \over 2} \end{align*}

(a)

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

(b)

\begin{align*} 2 \left( \begin{matrix} x \\ y \end{matrix} \right) - 4 \left( \begin{matrix} 2 \\ 5 \end{matrix} \right) & = \left( \begin{matrix} 4x \\ 3y \end{matrix} \right) \\ \left( \begin{matrix} 2x \\ 2y \end{matrix} \right) - \left( \begin{matrix} 8 \\ 20 \end{matrix} \right) & = \left( \begin{matrix} 4x \\ 3y \end{matrix} \right) \\ \left( \begin{matrix} 2x - 8 \\ 2y - 20 \end{matrix} \right) & = \left( \begin{matrix} 4x \\ 3y \end{matrix} \right) \end{align*} \begin{align*} 2x - 8 & = 4x &&& 2y - 20 & = 3y \\ 2x - 4x & = 8 &&& 2y - 3y & = 20 \\ -2x & = 8 &&& -y & = 20 \\ x & = -4 &&& y & = -20 \end{align*}

(c)

\begin{align*} x \left( \begin{matrix} 2 \\ 4 \end{matrix} \right) + y \left( \begin{matrix} 5 \\ 7 \end{matrix} \right) & = \left( \begin{matrix} -5 \\ -1 \end{matrix} \right) \\ \left( \begin{matrix} 2x \\ 4x \end{matrix} \right) + \left( \begin{matrix} 5y \\ 7y \end{matrix} \right) & = \left( \begin{matrix} -5 \\ -1 \end{matrix} \right) \\ \left( \begin{matrix} 2x + 5y \\ 4x + 7y \end{matrix} \right) & = \left( \begin{matrix} -5 \\ -1 \end{matrix} \right) \end{align*} \begin{align*} 2x + 5y & = -5 &&& 4x + 7y & = -1 \phantom{0} \text{--- (1)} \\ 2x & = -5 - 5y \\ x & = -2.5 - 2.5y \phantom{0} \text{--- (2)} \end{align*} \begin{align*} \text{Substitute } & \text{(2) into (1),} \\ 4(-2.5 - 2.5y) + 7y & = -1 \\ -10 - 10y + 7y & = -1 \\ -10y + 7y & = -1 + 10 \\ -3y & = 9 \\ y & = -3 \\ \\ \text{Substitute } & y = -3 \text{ into (2),} \\ x & = -2.5 - 2.5(-3) \\ x & = 5 \\ \\ \therefore x & = 5, y = -3 \end{align*}

(a)

\begin{align*} \underset{2 \times 2}{ \left( \begin{matrix} 0 & 2 \\ -1 & 4 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} x \\ y \end{matrix} \right) } & = \left( \begin{matrix} 2 \\ 3 \end{matrix} \right) \\ \underset{2 \times 1}{ \left( \begin{matrix} (0 \times x) + (2 \times y) \\ (-1 \times x) + (4 \times y) \end{matrix} \right) } & = \left( \begin{matrix} 2 \\ 3 \end{matrix} \right) \\ \left( \begin{matrix} 2y \\ -x + 4y \end{matrix} \right) & = \left( \begin{matrix} 2 \\ 3 \end{matrix} \right) \end{align*} \begin{align*} 2y & = 2 &&& -x + 4y & = 3 \\ y & = 1 &&& -x + 4(1) & = 3 \\ & &&& -x + 4 & = 3 \\ & &&& - x & = 3 - 4 \\ & &&& - x & = -1 \\ & &&& x & = 1 \end{align*}

(b)

\begin{align*} \underset{2 \times 2}{ \left( \begin{matrix} 2 & 1 \\ 4 & x \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} y \\ 2 \end{matrix} \right) } & = \left( \begin{matrix} 6 \\ 10 \end{matrix} \right) \\ \underset{2 \times 1}{ \left( \begin{matrix} (2 \times y) + (1 \times 2) \\ (4 \times y) + (x \times 2) \end{matrix} \right) } & = \left( \begin{matrix} 6 \\ 10 \end{matrix} \right) \\ \left( \begin{matrix} 2y + 2 \\ 4y + 2x \end{matrix} \right) & = \left( \begin{matrix} 6 \\ 10 \end{matrix} \right) \end{align*} \begin{align*} 2y + 2 & = 6 &&& 4y + 2x & = 10 \\ 2y & = 4 &&& 2x & = 10 - 4y \\ y & = 2 &&& 2x & = 10 - 4(2) \\ & &&& 2x & = 2 \\ & &&& x & = 1 \end{align*}

(c)

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

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

\begin{align*} \textbf{AB} & = \textbf{A} + \textbf{B} \\ \underset{2 \times 2}{ \left( \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} a & b \\ 0 & c \end{matrix} \right) } & = \left( \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} \right) + \left( \begin{matrix} a & b \\ 0 & c \end{matrix} \right) \\ \underset{2 \times 2}{ \left( \begin{matrix} (2 \times a) + (0 \times 0) & (2 \times b) + (0 \times c) \\ (0 \times a) + (3 \times 0) & (0 \times b) + (3 \times c) \end{matrix} \right) } & = \left( \begin{matrix} 2 + a & 0 \\ 0 & 3 + c \end{matrix} \right) \\ \left( \begin{matrix} 2a & 2b \\ 0 & 3c \end{matrix} \right) & = \left( \begin{matrix} 2 + a & 0 \\ 0 & 3 + c \end{matrix} \right) \end{align*} \begin{align*} 2a & = 2 + a &&& 2b & = 0 &&& 3c & = 3 + c \\ a & = 2 &&& b & = 0 &&& 2c & = 3 \\ & &&& & &&& c & = {3 \over 2} \end{align*}

(i)

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

(ii)

\begin{align*} \left( \begin{matrix} 3 & 1 \\ -2 & - 5 \end{matrix} \right) + \underset{2 \times 2}{ \left( \begin{matrix} -2 & 1 \\ -1 & 4 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} 3 & - 1 \\ 4 & - 2 \end{matrix} \right) } & = \left( \begin{matrix} 3 & 1 \\ -2 & - 5 \end{matrix} \right) + \underset{2 \times 2}{ \left( \begin{matrix} (-2 \times 3) + (1 \times 4) & (-2 \times -1) + (1 \times -2) \\ (-1 \times 3) + (4 \times 4) & (-1 \times -1) + (4 \times -2) \end{matrix} \right) } \\ & = \left( \begin{matrix} 3 & 1 \\ -2 & - 5 \end{matrix} \right) + \left( \begin{matrix} -2 & 0 \\ 13 & -7 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 & 1 \\ 11 & -12 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \left( \begin{matrix} -2 & 1 \\ -1 & 4 \end{matrix} \right) + \underset{2 \times 2}{ \left( \begin{matrix} 3 & 1 \\ -2 & - 5 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} 3 & - 1 \\ 4 & - 2 \end{matrix} \right) } & = \left( \begin{matrix} -2 & 1 \\ -1 & 4 \end{matrix} \right) + \underset{2 \times 2}{ \left( \begin{matrix} (3 \times 3) + (1 \times 4) & (3 \times -1) + (1 \times -2) \\ (-2 \times 3) + (-5 \times 4) & (-2 \times -1) + (-5 \times -2) \end{matrix} \right) } \\ & = \left( \begin{matrix} -2 & 1 \\ -1 & 4 \end{matrix} \right) + \left( \begin{matrix} 13 & -5 \\ -26 & 12 \end{matrix} \right) \\ & = \left( \begin{matrix} 11 & -4 \\ -27 & 16 \end{matrix} \right) \end{align*}

(a)

\begin{align*} \underset{2 \times 2}{ \left( \begin{matrix} 0 & 5 \\ -2 & 4 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} x \\ y \end{matrix} \right) } & = \left( \begin{matrix} -15 \\ 28 \end{matrix} \right) \\ \underset{2 \times 1}{ \left( \begin{matrix} (0 \times x) + (5 \times y) \\ (-2 \times x) + (4 \times y) \end{matrix} \right) } & = \left( \begin{matrix} -15 \\ 28 \end{matrix} \right) \\ \left( \begin{matrix} 5y \\ -2x + 4y \end{matrix} \right) & = \left( \begin{matrix} -15 \\ 28 \end{matrix} \right) \end{align*} \begin{align*} 5y & = -15 &&& -2x + 4y & = 28 \\ y & = -3 &&& -2x & = 28 - 4y \\ & &&& -2x & = 28 - 4(-3) \\ & &&& -2x & = 40 \\ & &&& x & = - 20 \end{align*}

(b)(i)

\begin{align*} \textbf{AB} & = \underset{2 \times 2}{ \left( \begin{matrix} 3 & 2 \\ 1 & 4 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} 1 \\ 3 \end{matrix} \right) } \\ & = \underset{2 \times 1}{ \left( \begin{matrix} (3 \times 1) + (2 \times 3) \\ (1 \times 1) + (4 \times 3) \end{matrix} \right) } \\ & = \left( \begin{matrix} 9 \\ 13 \end{matrix} \right) \end{align*}

(b)(ii)

\begin{align*} \textbf{AC} & = \underset{2 \times 2}{ \left( \begin{matrix} 3 & 2 \\ 1 & 4 \end{matrix} \right) } \underset{1 \times 2}{ \left( \begin{matrix} 2 & 1 \end{matrix} \right) } \\ \\ & \text{Not } \text{possible} \end{align*}

(b)(iii)

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

(b)(iv)

\begin{align*} \textbf{CB} & = \underset{1 \times 2}{ \left( \begin{matrix} 2 & 1 \end{matrix} \right) } \underset{2 \times 1}{ \left( \begin{matrix} 1 \\ 3 \end{matrix} \right) } \\ & = \underset{1 \times 1}{ \left( \begin{matrix} (2 \times 1) + (1 \times 3) \end{matrix} \right) } \\ & = \left( \begin{matrix} 5 \end{matrix} \right) \end{align*}

(a)

\begin{align*} \underset{2 \times 2}{ \left( \begin{matrix} 4 & 2 \\ -1 & p \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} 2 & 3 \\ 0 & 1 \end{matrix} \right) } & = \left( \begin{matrix} q & 14 \\ -2 & 7 \end{matrix} \right) \\ \underset{2 \times 2}{ \left( \begin{matrix} (4 \times 2) + (2 \times 0) & (4 \times 3) + (2 \times 1) \\ (-1 \times 2) + (p \times 0) & (-1 \times 3) + (p \times 1) \end{matrix} \right) } & = \left( \begin{matrix} q & 14 \\ -2 & 7 \end{matrix} \right) \\ \left( \begin{matrix} 8 & 14 \\ -2 & -3 + p \end{matrix} \right) & = \left( \begin{matrix} q & 14 \\ -2 & 7 \end{matrix} \right) \end{align*} \begin{align*} q & = 8 &&& -3 + p & = 7 \\ & &&& p & = 10 \end{align*}

(b)(i)

\begin{align*} \textbf{AB} & = \underset{2 \times 2}{ \left( \begin{matrix} 3 & 1 \\ 0 & 2 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} 2 & 6 \\ 0 & k \end{matrix} \right) } \\ & = \underset{2 \times 2}{ \left( \begin{matrix} (3 \times 2) + (1 \times 0) & (3 \times 6) + (1 \times k) \\ (0 \times 2) + (2 \times 0) & (0 \times 6) + (2 \times k) \end{matrix} \right) } \\ & = \left( \begin{matrix} 6 & 18 + k \\ 0 & 2k \end{matrix} \right) \\ \\ \textbf{BA} & = \underset{2 \times 2}{ \left( \begin{matrix} 2 & 6 \\ 0 & k \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} 3 & 1 \\ 0 & 2 \end{matrix} \right) } \\ & = \underset{2 \times 2}{ \left( \begin{matrix} (2 \times 3) + (6 \times 0) & (2 \times 1) + (6 \times 2) \\ (0 \times 3) + (k \times 0) & (0 \times 1) + (k \times 2) \end{matrix} \right) } \\ & = \left( \begin{matrix} 6 & 14 \\ 0 & 2k \end{matrix} \right) \end{align*}

(b)(ii)

\begin{align*} \textbf{AB} & = \textbf{BA} \\ \left( \begin{matrix} 6 & 18 + k \\ 0 & 2k \end{matrix} \right) & = \left( \begin{matrix} 6 & 14 \\ 0 & 2k \end{matrix} \right) \\ \\ 18 + k & = 14 \\ k & = -4 \end{align*}

(i)

\begin{align*} \textbf{R} & = \textbf{AM} \\ & = \underset{2 \times 3}{ \left( \begin{matrix} 39 & 5 & 6 \\ 29 & 8 & 13 \end{matrix} \right) } \underset{3 \times 1}{ \left( \begin{matrix} 2 \\ 0 \\ -1 \end{matrix} \right) } \\ & = \underset{2 \times 1}{ \left( \begin{matrix} (39 \times 2) + (5 \times 0) + (6 \times -1) \\ (29 \times 2) + (8 \times 0) + (13 \times -1) \end{matrix} \right) } \\ & = \left( \begin{matrix} 72 \\ 45 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{R} & = \left( \begin{matrix} 72 \\ 45 \end{matrix} \right) \begin{matrix} \text{Li Ting's marks} \\ \text{Bernard's marks} \end{matrix} \\ \\ \text{The marks obtained} & \text{ by Li Ting and Bernard } \textbf{respectively}. \end{align*}

\begin{align*} \textbf{AB} & = \textbf{BA} \\ \underset{2 \times 2}{ \left( \begin{matrix} -4 & p \\ -1 & 2 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} q & 0 \\ 2 & 3 \end{matrix} \right) } & = \underset{2 \times 2}{ \left( \begin{matrix} q & 0 \\ 2 & 3 \end{matrix} \right) } \underset{2 \times 2}{ \left( \begin{matrix} -4 & p \\ -1 & 2 \end{matrix} \right) } \\ \underset{2 \times 2}{ \left( \begin{matrix} (-4 \times q) + (p \times 2) & (-4 \times 0) + (p \times 3) \\ (-1 \times q) + (2 \times 2) & (-1 \times 0) + (2 \times 3) \end{matrix} \right) } & = \underset{2 \times 2}{ \left( \begin{matrix} (q \times -4) + (0 \times -1) & (q \times p) + (0 \times 2) \\ (2 \times -4) + (3 \times -1) & (2 \times p) + (3 \times 2) \end{matrix} \right) } \\ \left( \begin{matrix} -4q + 2p & 3p \\ -q + 4 & 6 \end{matrix} \right) & = \left( \begin{matrix} -4q & pq \\ -11 & 2p + 6 \end{matrix} \right) \end{align*} \begin{align*} -4q + 2p & = -4q &&& -q + 4 & = -11 \\ 2p & = -4q + 4q &&& -q & = -15 \\ 2p & = 0 &&& q & = 15 \\ p & = 0 \end{align*}

(i)

\begin{align*} 12 \textbf{A} & = 12 \left( \begin{matrix} 120 \\ 95 \\ 102 \end{matrix} \right) \\ & = \left( \begin{matrix} 1440 \\ 1140 \\ 1224 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{B} & = 12 \textbf{A} + \textbf{W} \\ & = \left( \begin{matrix} 1440 \\ 1140 \\ 1224 \end{matrix} \right) + \left( \begin{matrix} 60 \\ 75 \\ 62 \end{matrix} \right) \\ & = \left( \begin{matrix} 1500 \\ 1215 \\ 1286 \end{matrix} \right) \\ \\ \textbf{B} \text{ represents the total cost for 1 year} & \text{ and additional World Cup package for each cable provider respectively} \end{align*}

(i)

\begin{align*} \textbf{P} & = \left( \begin{matrix} 3 \\ 1 \\ 0 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{RP} & = \underset{3 \times 3} { \left( \begin{matrix} 12 & 5 & 3 \\ 3 & 8 & 7 \\ 9 & 4 & 4 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 3 \\ 1 \\ 0 \end{matrix} \right) } \\ & = \underset{3 \times 1} { \left( \begin{matrix} (12 \times 3) + (5 \times 1) + (3 \times 0) \\ (3 \times 3) + (8 \times 1) + (7 \times 0) \\ (9 \times 3) + (4 \times 1) + (4 \times 0) \end{matrix} \right) } \\ & = \left( \begin{matrix} 41 \\ 17 \\ 31 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \textbf{RP}\text{ represents the total points the three teams scored respectively.} \end{align*}

(i)

\begin{align*} \textbf{K} & = 5 \textbf{A} \\ & = 5 \left( \begin{matrix} 13 & 14 \\ 10 & 12 \end{matrix} \right) \\ & = \left( \begin{matrix} 65 & 70 \\ 50 & 60 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{L} & = \underset{2 \times 2} { \left( \begin{matrix} 65 & 70 \\ 50 & 60 \end{matrix} \right) } \underset{2 \times 1} { \left( \begin{matrix} 2.50 \\ 1.80 \end{matrix} \right) } \\ & = \underset{2 \times 1} { \left( \begin{matrix} (65 \times 2.50) + (70 \times 1.80) \\ (50 \times 2.50) + (60 \times 1.80) \end{matrix} \right) } \\ & = \left( \begin{matrix} 288.5 \\ 233 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \text{They represent the total fees collected from boys and girls respectively.} \end{align*}

(i)

\begin{align*} \textbf{PQ} & = \underset{2 \times 3} { \left( \begin{matrix} 75 & 84 & 135 \\ 88 & 95 & 140 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 0.70 \\ 0.85 \\ 1 \end{matrix} \right) } \\ & = \underset{2 \times 1} { \left( \begin{matrix} (75 \times 0.7) + (85 \times 0.85) + (135 \times 1) \\ (88 \times 0.7) + (95 \times 0.85) + (140 \times 1) \end{matrix} \right) } \\ & = \left( \begin{matrix} 258.9 \\ 282.35 \end{matrix} \right) \\ \\ \text{Total cost of } & \text{beverages sold for Cafe A and Cafe B respectively} \end{align*}

(ii)

\begin{align*} \textbf{R} - \textbf{Q} & = \left( \begin{matrix} 3.20 \\ 4 \\ 4.50 \end{matrix} \right) - \left( \begin{matrix} 0.70 \\ 0.85 \\ 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 2.50 \\ 3.15 \\ 3.50 \end{matrix} \right) \\ \\ \text{Profit of a cup of latte} & \text{, a cup of mocha and a cup of iced lemon tea respectively.} \end{align*}

(iii)

\begin{align*} \textbf{P} (\textbf{R} - \textbf{Q}) & = \underset{2 \times 3} { \left( \begin{matrix} 75 & 84 & 135 \\ 88 & 95 & 140 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 2.50 \\ 3.15 \\ 3.50 \end{matrix} \right) } \\ & = \underset{2 \times 1} { \left( \begin{matrix} (75 \times 2.5) + (84 \times 3.15) + (135 \times 3.5) \\ (88 \times 2.5) + (95 \times 3.15) + (140 \times 3.5) \end{matrix} \right) } \\ & = \left( \begin{matrix} 924.6 \\ 1 \phantom{.} 009.25 \end{matrix} \right) \\ \\ \text{Profit earned by } & \text{Cafe A and Cafe B on the particular day respectively.} \end{align*}

(i)

\begin{align*} \underset{3 \times 4} { \left( \begin{matrix} 50 & 60 & 70 & 40 \\ 30 & 40 & 50 & 30 \\ 40 & 30 & 60 & 50 \end{matrix} \right) } \underset{4 \times 1} { \left( \begin{matrix} 3.2 \\ 3.1 \\ 3 \\ 3.3 \end{matrix} \right) } & = \underset{3 \times 1} { \left( \begin{matrix} (50 \times 3.2) + (60 \times 3.1) + (70 \times 3) + (40 \times 3.3) \\ (30 \times 3.2) + (40 \times 3.1) + (50 \times 3) + (30 \times 3.3) \\ (40 \times 3.2) + (30 \times 3.1) + (60 \times 3) + (50 \times 3.3) \end{matrix} \right) } \\ & = \left( \begin{matrix} 688 \\ 469 \\ 566 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} & \phantom{=0} \underset{1 \times 3} { \left( \begin{matrix} 25 & 12 & 15 \end{matrix} \right) } \underset{3 \times 4} { \left( \begin{matrix} 50 & 60 & 70 & 40 \\ 30 & 40 & 50 & 30 \\ 40 & 30 & 60 & 50 \end{matrix} \right) } \\ & = \underset{1 \times 4} { \left( \begin{matrix} (25 \times 50) + (12 \times 30) + (15 \times 40) & (25 \times 60) + (12 \times 40) + (15 \times 30) & (25 \times 70) + (12 \times 50) + (15 \times 60) & (25 \times 40) + (12 \times 30) + (15 \times 50) \end{matrix} \right) } \\ & = \left( \begin{matrix} 2210 & 2430 & 3250 & 2110 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \text{Total no. of tubs} & = 2210 + 2430 + 3250 + 2110 \\ & = 10 \phantom{.} 000 \end{align*}

(i)

\begin{align*} \textbf{X} & = \underset{3 \times 3} { \left( \begin{matrix} 65 & 45 & 46 \\ 60 & 56 & 58 \\ 78 & 56 & 54 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 10 \\ 20 \\ 50 \end{matrix} \right) } \\ & = \underset{3 \times 1} { \left( \begin{matrix} (65 \times 10) + (45 \times 20) + (46 \times 50) \\ (60 \times 10) + (56 \times 20) + (58 \times 50) \\ (78 \times 10) + (56 \times 20) + (54 \times 50) \end{matrix} \right) } \\ & = \left( \begin{matrix} 3850 \\ 4620 \\ 4600 \end{matrix} \right) \\ \\ \text{Total value of } & \text{10 cents, 20 cents and 50 cents coins collected respectively.} \end{align*}

(ii)

\begin{align*} \textbf{Y} & = \underset{3 \times 4} { \left( \begin{matrix} 32 & 26 & 18 & 16 \\ 45 & 34 & 20 & 10 \\ 38 & 22 & 25 & 24 \end{matrix} \right) } \underset{4 \times 1} { \left( \begin{matrix} 1 \\ 2 \\ 5 \\ 10 \end{matrix} \right) } \\ & = \underset{3 \times 1} { \left( \begin{matrix} (32 \times 1) + (26 \times 2) + (18 \times 5) + (16 \times 10) \\ (45 \times 1) + (34 \times 2) + (20 \times 5) + (10 \times 10) \\ (38 \times 1) + (22 \times 2) + (25 \times 5) + (24 \times 10) \end{matrix} \right) } \\ & = \left( \begin{matrix} 334 \\ 313 \\ 447 \end{matrix} \right) \\ \\ \text{Total value of } & \text{1 dollar, 2 dollars, 5 dollars and 10 dollars notes collected respectively.} \end{align*}

(iii)

\begin{align*} \textbf{S} & = {1 \over 100} \underset{1 \times 3} { \left( \begin{matrix} 1 & 1 & 1 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 3850 \\ 4620 \\ 4600 \end{matrix} \right) } \\ & = {1 \over 100} \underset{1 \times 1} { \left( \begin{matrix} 3850 + 4620 + 4600 \end{matrix} \right) } \\ & = {1 \over 100} \left( \begin{matrix} 13 \phantom{.} 070 \end{matrix} \right) \\ & = \left( \begin{matrix} 130.7 \end{matrix} \right) \\ \\ \text{The total value of } & \text{10 cents, 20 cents and 50 cents coins collected in dollars.} \end{align*}

(iv)

\begin{align*} \textbf{T} & = \underset{1 \times 3} { \left( \begin{matrix} 1 & 1 & 1 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 334 \\ 313 \\ 447 \end{matrix} \right) } + \left( \begin{matrix} 130.7 \end{matrix} \right) \\ & = \underset{1 \times 1} { \left( \begin{matrix} 334 + 313 + 447 \end{matrix} \right) } + \left( \begin{matrix} 130.7 \end{matrix} \right) \\ & = \left( \begin{matrix} 1094 \end{matrix} \right) + \left( \begin{matrix} 130.7 \end{matrix} \right) \\ & = \left( \begin{matrix} 1224.7 \end{matrix} \right) \\ \\ \text{The total } & \text{amount collected from the charity event} \end{align*}

(i)

\begin{align*} \underset{3 \times 3} { \left( \begin{matrix} 12 & 8 & 11 \\ 18 & 11 & 7 \\ 8 & 9 & 15 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 48 \\ 32 \\ 26 \end{matrix} \right) } & = \underset{3 \times 1} { \left( \begin{matrix} (12 \times 48) + (8 \times 32) + (11 \times 26) \\ (18 \times 48) + (11 \times 32) + (7 \times 26) \\ (8 \times 48) + (9 \times 32) + (15 \times 26) \end{matrix} \right) } \begin{matrix} \text{Robert} \\ \text{Star} \\ \text{Mido} \end{matrix} \\ & = \left( \begin{matrix} 1118 \\ 1398 \\ 1062 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \underset{1 \times 3} { \left( \begin{matrix} 9.8 & 10.4 & 9.9 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 1118 \\ 1398 \\ 1062 \end{matrix} \right) } & = \underset{1 \times 1} { \left( \begin{matrix} (9.8 \times 1118) + (10.4 \times 1398) + (9.9 \times 1062) \end{matrix} \right) } \\ & = \left( \begin{matrix} 36 \phantom{.} 009.4 \end{matrix} \right) \end{align*}

(i)

\begin{align*} \text{HK\$} 100 & = \text{S\$} 16 \\ \text{HK\$} 1 & = \text{S\$} {16 \over 100} \\ \\ \text{RM100} & = \text{S\$}40 \\ \text{RM1} & = \text{S\$} {40 \over 100} \\ \\ \underset{3 \times 2} { \left( \begin{matrix} 125 & 320 \\ 200 & 160 \\ 90 & 450 \end{matrix} \right) } \underset{2 \times 1} { \left( \begin{matrix} {16 \over 100} \\ {40 \over 100} \end{matrix} \right) } & = \underset{3 \times 1} { \left( \begin{matrix} (125 \times {16 \over 100}) + (320 \times {40 \over 100}) \\ (200 \times {16 \over 100}) + (160 \times {40 \over 100}) \\ (90 \times {16 \over 100}) + (450 \times {40 \over 100}) \end{matrix} \right) } \\ & = \left( \begin{matrix} 148 \\ 96 \\ 194.4 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \underset{1 \times 3} { \left( \begin{matrix} 1 & 1 & 1 \end{matrix} \right) } \underset{3 \times 1} { \left( \begin{matrix} 148 \\ 96 \\ 194.4 \end{matrix} \right) } & = \underset{1 \times 1} { \left( \begin{matrix} 148 + 96 + 194.4 \end{matrix} \right) } \\ & = \left( \begin{matrix} 438.4 \end{matrix} \right) \end{align*}

(a)

\begin{align*} \textbf{Q} & = \left( \begin{matrix} 4 & 3 & 2 \\ 4 & 4 & 0 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \textbf{R} & = \underset{2 \times 3} { \left( \begin{matrix} 4 & 3 & 2 \\ 4 & 4 & 0 \end{matrix} \right) } \underset{3 \times 2} { \left( \begin{matrix} 42 & -3 \\ 48 & 5 \\ 56 & -4 \end{matrix} \right) } \\ & = \underset{2 \times 2} { \left( \begin{matrix} (4 \times 42) + (3 \times 48) + (2 \times 56) & (4 \times -3) + (3 \times 5) + (2 \times -4) \\ (4 \times 42) + (4 \times 48) + (0 \times 56) & (4 \times -3) + (4 \times 5) + (0 \times -4) \end{matrix} \right) } \\ & = \left( \begin{matrix} 424 & -5 \\ 360 & 8 \end{matrix} \right) \begin{matrix} \text{Siti} \\ \text{Vani} \end{matrix} \end{align*}

(c)

\begin{align*} \textbf{R} & = \left( \begin{matrix} 424 & -5 \\ 360 & 8 \end{matrix} \right) \begin{matrix} \text{Siti} \\ \text{Vani} \end{matrix} \\ \\ \text{Siti should } & \text{buy from Bakery B as it is \$ 5 cheaper} \end{align*}

(d)

\begin{align*} \textbf{RS} & = \underset{2 \times 2} { \left( \begin{matrix} 424 & -5 \\ 360 & 8 \end{matrix} \right) } \underset{2 \times 1} { \left( \begin{matrix} 1 \\ 1 \end{matrix} \right) } \\ \\ \textbf{S} & = \left( \begin{matrix} 1 \\ 1 \end{matrix} \right) \end{align*}