(a)

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

(b)

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

(c)

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

(d)

\begin{align*} {1 \over 3} \left( \begin{matrix} 6 & 15 \\ 21 & -24 \end{matrix} \right) & = \left( \begin{matrix} 2 & 5 \\ 7 & -8 \end{matrix} \right) \end{align*}

(e)

\begin{align*} -2 \left( \begin{matrix} -1 & 0.5 & 3 \\ -0.8 & 2 & 1.2 \end{matrix} \right) & = \left( \begin{matrix} 2 & -1 & -6 \\ 1.6 & -4 & -2.4 \end{matrix} \right) \end{align*}

(f)

\begin{align*} 5 \left( \begin{matrix} 1 & 5 \\ -4 & 3 \\ -1 & 2 \end{matrix} \right) & = \left( \begin{matrix} 5 & 25 \\ -20 & 15 \\ -5 & 10 \end{matrix} \right) \end{align*}

(g)

\begin{align*} 3 \left( \begin{matrix} 6 & { 1 \over 2} & 1 \\ 0 & 2 & {1 \over 3} \\ 5 & -4 & -2 \end{matrix} \right) & = \left( \begin{matrix} 18 & {3 \over 2} & 3 \\ 0 & 6 & 1 \\ 15 & -12 & -6 \end{matrix} \right) \end{align*}

(a)

\begin{align*} 2 \left( \begin{matrix} -1 \\ 5 \end{matrix} \right) + 3 \left( \begin{matrix} 4 \\ -3 \end{matrix} \right) & = \left( \begin{matrix} -2 \\ 10 \end{matrix} \right) + \left( \begin{matrix} 12 \\ -9 \end{matrix} \right) \\ & = \left( \begin{matrix} 10 \\ 1 \end{matrix} \right) \end{align*}

(b)

\begin{align*} 2\left( \begin{matrix} 3 & 1 & 5 \end{matrix} \right) - 4 \left( \begin{matrix} -1 & 3 & 2 \end{matrix} \right) & = \left( \begin{matrix} 6 & 2 & 10 \end{matrix} \right) - \left( \begin{matrix} -4 & 12 & 8 \end{matrix} \right) \\ & = \left( \begin{matrix} 10 & -10 & 2 \end{matrix} \right) \end{align*}

(c)

\begin{align*} 5 \left( \begin{matrix} 1 & 3 \\ -4 & 6 \end{matrix} \right) - 2 \left( \begin{matrix} -3 & -1 \\ 4 & 2 \end{matrix} \right) & = \left( \begin{matrix} 5 & 15 \\ -20 & 30 \end{matrix} \right) - \left( \begin{matrix} -6 & -2 \\ 8 & 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 11 & 17 \\ -28 & 26 \end{matrix} \right) \end{align*}

(d)

\begin{align*} 3 \left( \begin{matrix} 0 & 4 & 1 \\ 5 & 0 & -1 \end{matrix} \right) - 4 \left( \begin{matrix} -1 & 3 & 0 \\ -2 & 1 & -1 \end{matrix} \right) & = \left( \begin{matrix} 0 & 12 & 3 \\ 15 & 0 & -3 \end{matrix} \right) - \left( \begin{matrix} -4 & 12 & 0 \\ -8 & 4 & -4 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 & 0 & 3 \\ 23 & -4 & 1 \end{matrix} \right) \end{align*}

(i)

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

(ii)

\begin{align*} \textbf{A} + 2\textbf{B} & = \left( \begin{matrix} 4 & 4 \\ 2 & 7 \end{matrix} \right) + 2\left( \begin{matrix} 1 & 2 \\ -1 & 3 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 & 4 \\ 2 & 7 \end{matrix} \right) + \left( \begin{matrix} 2 & 4 \\ -2 & 6 \end{matrix} \right) \\ & = \left( \begin{matrix} 6 & 8 \\ 0 & 13 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \textbf{A} - \textbf{B} - \textbf{C} & = \left( \begin{matrix} 4 & 4 \\ 2 & 7 \end{matrix} \right) - \left( \begin{matrix} 1 & 2 \\ -1 & 3 \end{matrix} \right) - \left( \begin{matrix} 1 & 4 \\ 3 & -5\end{matrix} \right) \\ & = \left( \begin{matrix} 3 & 2 \\ 3 & 4 \end{matrix} \right) - \left( \begin{matrix} 1 & 4 \\ 3 & - 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 2 & -2 \\ 0 & 9 \end{matrix} \right) \end{align*}

(iv)

\begin{align*} 2\textbf{A} - 2\textbf{C} + 3\textbf{C} & = 2\left( \begin{matrix} 4 & 4 \\ 2 & 7 \end{matrix} \right) - 2\left( \begin{matrix} 1 & 4 \\ 3 & -5\end{matrix} \right) + 3\left( \begin{matrix} 1 & 2 \\ -1 & 3 \end{matrix} \right) \\ & = \left( \begin{matrix} 8 & 8 \\ 4 & 14 \end{matrix} \right)- \left( \begin{matrix} 2 & 8 \\ 6 & -10 \end{matrix} \right) + \left( \begin{matrix} 3 & 6 \\ -3 & 9 \end{matrix} \right) \\ & = \left( \begin{matrix} 6 & 0 \\ -2 & 24 \end{matrix} \right) + \left( \begin{matrix} 3 & 6 \\ -3 & 9 \end{matrix} \right) \\ & = \left( \begin{matrix} 9 & 6 \\ -5 & 33 \end{matrix} \right) \end{align*}

(a)

\begin{align*} a\left( \begin{matrix} 2 \\ 2 \end{matrix} \right) + b\left( \begin{matrix} 2 \\ -2 \end{matrix} \right) & = \left( \begin{matrix} 0 \\ 8 \end{matrix} \right) \\ \left( \begin{matrix} 2a \\ 2a \end{matrix} \right) + \left( \begin{matrix} 2b \\ -2b \end{matrix} \right) & = \left( \begin{matrix} 0 \\ 8 \end{matrix} \right) \\ \left( \begin{matrix} 2a + 2b \\ 2a-2b \end{matrix} \right) & = \left( \begin{matrix} 0 \\ 8 \end{matrix} \right) \\ \\ 2a + 2b & = 0 \phantom{000} \text{ --- (1)} \\ 2a - 2b & = 8 \phantom{000} \text{ --- (2)} \\ \\ (1) & + (2), \\ \\ (2a + 2b) + (2a - 2b) & = 0 + 8 \\ 2a + 2b + 2a - 2b & = 8 \\ 4a & = 8 \\ a & = {8 \over 4} \\ a & = 2 \\ \\ \text{Substitute } & a = 2 \text{ into (1),} \\ 2(2) + 2b & = 0 \\ 4 + 2b & = 0 \\ 2b & = -4 \\ b & = {-4 \over 2} \\ b & = -2 \end{align*}

(b)

\begin{align*} 3\left( \begin{matrix} 2x \\ y \end{matrix} \right) + 3\left( \begin{matrix} x \\ 3y \end{matrix} \right) & = \left( \begin{matrix} 18 \\ 36 \end{matrix} \right) \\ \left( \begin{matrix} 6x \\ 3y \end{matrix} \right) + \left( \begin{matrix} 3x \\ 9y \end{matrix} \right) & = \left( \begin{matrix} 18 \\ 36 \end{matrix} \right) \\ \left( \begin{matrix} 9x \\ 12y \end{matrix} \right) & = \left( \begin{matrix} 18 \\ 36 \end{matrix} \right) \\ \\ 9x & = 18 \\ x & = {18 \over 9} \\ x & = 2 \\ \\ 12y & = 36 \\ y & = {36 \over 12} \\ y & = 3 \end{align*}

(c)

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

(d)

\begin{align*} 2\left( \begin{matrix} 5 & 3 & 2 \\ 1 & 6 & 3 \end{matrix} \right) + \left( \begin{matrix} a & b & c \\ -2 & -4 & 5 \end{matrix} \right) & = \left( \begin{matrix} 9 & 12 & 6 \\ d & e & f \end{matrix} \right) \\ \left( \begin{matrix} 10 & 6 & 4 \\ 2 & 12 & 6 \end{matrix} \right) + \left( \begin{matrix} a & b & c \\ -2 & - 4 & 5 \end{matrix} \right) & = \left( \begin{matrix} 9 & 12 & 6 \\ d & e & f \end{matrix} \right) \\ \left( \begin{matrix} 10 + a & 6 + b & 4 + c \\ 0 & 8 & 11 \end{matrix} \right) & = \left( \begin{matrix} 9 & 12 & 6 \\ d & e & f \end{matrix} \right) \\ \\ 10 + a & = 9 \\ a & = 9 - 10 \\ a & = -1 \\ \\ 6 + b & = 12 \\ b & = 12 - 6 \\ b & = 6 \\ \\ 4 + c & = 6 \\ c & = 6 - 4 \\ c & = 2 \\ \\ d & = 0 \\ \\ e & = 8 \\ \\ f & = 11 \end{align*}

(i)

\begin{align*} \text{Annual fees in dollars} & = 12 \textbf{C} \\ & = 12 \left( \begin{matrix} 680 \\ 720 \\ 635 \end{matrix} \right) \\ & = \left( \begin{matrix} 8160 \\ 8640 \\ 7620 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} 12 \textbf{C} + \textbf{J} + \textbf{D} & = \left( \begin{matrix} 8160 \\ 8640 \\ 7620 \end{matrix} \right) + \left( \begin{matrix} 150 \\ 120 \\ 200 \end{matrix} \right) + \left( \begin{matrix} 180 \\ 150 \\ 200 \end{matrix} \right) \\ & = \left( \begin{matrix} 8490 \\ 8910 \\ 8020 \end{matrix} \right) \\ \\ \\ \text{It represents } & \text{the total fees (inclusive of special programmes)} \\ \text{charged by e} & \text{ach centre for the entire year} \end{align*}

(iii)

\begin{align*} 12\textbf{C} + \textbf{J} & = \left( \begin{matrix} 8160 \\ 8640 \\ 7620 \end{matrix} \right) + \left( \begin{matrix} 150 \\ 120 \\ 200 \end{matrix} \right) \\ & = \left( \begin{matrix} 8310 \\ 8760 \\ 7820 \end{matrix} \right) \\ \\ \\ \text{Childcare} & \text{ centre } Z \text{ charges the lowest} \end{align*}

(a)

\begin{align*} \underset{2 \times 2}{\left( \begin{matrix} 4 & 3 \\ -1 & 5 \end{matrix} \right)} \underset{2 \times 1}{\left( \begin{matrix} -2 \\ 6 \end{matrix} \right)} & = \left( \begin{matrix} 4\times (-2) + 3 \times 6 \\ (-1) \times (-2) + 5 \times 6 \end{matrix} \right) \\ & = \left( \begin{matrix} 10 \\ 32 \end{matrix} \right) \end{align*}

(b)

\begin{align*} \underset{2 \times 2}{\left( \begin{matrix} -3 & 1 \\ 0 & 8 \end{matrix} \right)} \underset{2 \times 2}{\left( \begin{matrix} 6 & 5 \\ 8 & 7 \end{matrix} \right)} & = \left( \begin{matrix} (-3) \times 6 + 1 \times 8 & (-3) \times 5 + 1 \times 7 \\ 0 \times 6 + 8 \times 8 & 0 \times 5 + 8 \times 7 \end{matrix} \right) \\ & = \left( \begin{matrix} -10 & -8 \\ 64 & 56 \end{matrix} \right) \end{align*}

(c)

\begin{align*} \underset{2 \times 1}{\left( \begin{matrix} 6 \\ 7 \end{matrix} \right)} \underset{1 \times 2}{\left( \begin{matrix} -1 & 3 \end{matrix} \right)} & = \left( \begin{matrix} 6 \times (-1) & 6 \times 3 \\ 7 \times (-1) & 7 \times 3 \end{matrix} \right) \\ & = \left( \begin{matrix} -6 & 18 \\ -7 & 21 \end{matrix} \right) \end{align*}

(d)

\begin{align*} \underset{1 \times 2}{\left( \begin{matrix} -1 & 3 \end{matrix} \right)} \underset{2 \times 1}{\left( \begin{matrix} 6 \\ 7 \end{matrix} \right)} & = \left( \begin{matrix} (-1) \times 6 + 3 \times 7 \end{matrix} \right) \\ & = \left( \begin{matrix} 15 \end{matrix} \right) \end{align*}

(e)

\begin{align*} & \underset{2 \times 1}{\left( \begin{matrix} 1 \\ 8 \end{matrix} \right)} \underset{2 \times 2}{\left( \begin{matrix} 2 & 3 \\ -11 & 20 \end{matrix} \right)} \\ \\ & \text{Not } \text{possible} \end{align*}

(f)

\begin{align*} \underset{3 \times 2}{\left( \begin{matrix} -1 & 2 \\ 8 & 5 \\ 3 & -7 \end{matrix} \right)} \underset{2 \times 2}{\left( \begin{matrix} 2 & 3 \\ -1 & 4 \end{matrix} \right)} & = \left( \begin{matrix} (-1) \times 2 + 2 \times (-1) & (-1) \times 3 + 2 \times 4 \\ 8 \times 2 + 5 \times (-1) & 8 \times 3 + 5 \times 4 \\ 3 \times 2 + (-7) \times (-1) & 3 \times 3 + (-7) \times 4 \end{matrix} \right) \\ & = \left( \begin{matrix} -4 & 5 \\ 11 & 44 \\ 13 & -19 \end{matrix} \right) \end{align*}

(g)

\begin{align*} \underset{3 \times 4}{\left( \begin{matrix} 3 & 8 & 0 & 5 \\ -1 & 0 & 7 & 6 \\ 4 & 9 & -2 & 1 \end{matrix} \right)} \underset{4 \times 1}{\left( \begin{matrix} 2 \\ -3 \\ 0 \\ 5 \end{matrix} \right)} & = \left( \begin{matrix} 3 \times 2 + 8 \times (-3) + 0 \times 0 + 5 \times 5 \\ (-1) \times 2 + 0 \times (-3) + 7 \times 0 + 6 \times 5 \\ 4 \times 2 + 9 \times (-3) + (-2) \times 0 + 1 \times 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 7 \\ 28 \\ -14 \end{matrix} \right) \end{align*}

(h)

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

\begin{align*} \left( \begin{matrix} 1 & 5 \\ 3 & p \end{matrix} \right) \left( \begin{matrix} q \\ 7 \end{matrix} \right) & = \left( \begin{matrix} 50 \\ 35 \end{matrix} \right) \\ \left( \begin{matrix} 1 \times q + 5 \times 7 \\ 3 \times q + p \times 7 \end{matrix} \right) & = \left( \begin{matrix} 50 \\ 35 \end{matrix} \right) \\ \left( \begin{matrix} q + 35 \\ 3q + 7p \end{matrix} \right) & = \left( \begin{matrix} 50 \\ 35 \end{matrix} \right) \\ \\ q + 35 & = 50 \\ q & = 50 - 35 \\ q & = 15 \\ \\ 3q + 7p & = 35 \\ 3(15) + 7p & = 35 \\ 45 + 7p & = 35 \\ 7p & = 35 - 45 \\ 7p & = -10 \\ p & = -{10 \over 7} \end{align*}

(i)

\begin{align*} \textbf{AB} & = \left( \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right) \left( \begin{matrix} 1 & 0 \\ 2 & k \end{matrix} \right) \\ & = \left( \begin{matrix} 2 \times 1 + 0 \times 2 & 2 \times 0 + 0 \times k \\ 1 \times 1 + 5 \times 2 & 1 \times 0 + 5 \times k \end{matrix} \right) \\ & = \left( \begin{matrix} 2 & 0 \\ 11 & 5k \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{BA} & = \left( \begin{matrix} 1 & 0 \\ 2 & k \end{matrix} \right) \left( \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 \times 2 + 0 \times 1 & 1 \times 0 + 0 \times 5 \\ 2 \times 2 + k \times 1 & 2 \times 0 + k \times 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 2 & 0 \\ 4 + k & 5k \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \textbf{AB} & = \textbf{BA} \\ \left( \begin{matrix} 2 & 0 \\ 11 & 5k \end{matrix} \right) & = \left( \begin{matrix} 2 & 0 \\ 4 + k & 5k \end{matrix} \right) \\ \\ 11 & = 4 + k \\ 11 - 4 & = k \\ 7 & = k \end{align*}

(i)

\begin{align*} \textbf{AI} & = \left( \begin{matrix} 8 & -3 \\ 7 & 5 \end{matrix} \right) \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 8 \times 1 + (-3) \times 0 & 8 \times 0 + (-3) \times 1 \\ 7 \times 1 + 5 \times 0 & 7 \times 0 + 5 \times 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 8 & -3 \\ 7 & 5 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{IA} & = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \left( \begin{matrix} 8 & -3 \\ 7 & 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 \times 8 + 0 \times 7 & 1 \times (-3) + 0 \times 5 \\ 0 \times 8 + 1 \times 7 & 0 \times (-3) + 1 \times 5 \end{matrix} \right) \\ & = \left( \begin{matrix} 8 & -3 \\ 7 & 5 \end{matrix} \right) \\ \\ \\ \text{Yes, } & \textbf{AI} = \textbf{A} = \textbf{IA} \end{align*}

\begin{align*} \left( \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix} \right) & = \left( \begin{matrix} 7 & 0 \\ 0 & 0 \end{matrix} \right) + \left( \begin{matrix} 0 & 6 \\ 0 & 0 \end{matrix} \right) + \left( \begin{matrix} 0 & 0 \\ 4 & 0 \end{matrix} \right) + \left( \begin{matrix} 0 & 0 \\ 0 & 3 \end{matrix} \right) \\ & = 7\textbf{A} + 6\textbf{B} + 4 \textbf{C} + 3 \textbf{D} \end{align*}

(i)

\begin{align*} \textbf{A} & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \\ \textbf{B} & = \left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right) \\ \\ \textbf{AB} & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 + 2 & 1 + 2 \\ 3 + 4 & 3 + 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 3 & 3 \\ 7 & 7 \end{matrix} \right) \\ \\ \textbf{BA} & = \left( \begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix} \right) \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 + 3 & 2 + 4 \\ 1 + 3 & 2 + 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 4 & 6 \\ 4 & 6 \end{matrix} \right) \end{align*}

(ii)

\begin{align*} \textbf{A} & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \\ \textbf{B} & = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \\ \\ \textbf{AB} & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 + 0 & 0 + 2 \\ 3 + 0 & 0 + 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \\ \\ \textbf{BA} & = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 + 0 & 2 + 0 \\ 0 + 3 & 0 + 4 \end{matrix} \right) \\ & = \left( \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right) \end{align*}

(iii)

\begin{align*} \textbf{A}^2 & = \textbf{A} \times \textbf{A} \\ & = \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \\ & = \left( \begin{matrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{matrix} \right) \\ \\ \\ \therefore \textbf{A}^2 & \ne \left( \begin{matrix} a^2 & b^2 \\ c^2 & d^2 \end{matrix} \right) \end{align*}