Matrix multiplication

1. Given that

$$\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 3 \\ - 2 & 3 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & k \\ 0 & - 3 \\ 2 k & 2 \end{array} \right)$$ where \(k\) is a non-zero constant,

1. determine the matrix \(\mathbf { A B }\)
2. determine the value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)

4 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { l l l } 2 & - 1 & 5 \end{array} \right)$$ and $$\mathbf { E } = \mathbf { C D }$$ find \(\mathbf { E }\).\\ 3. $$f ( x ) = \frac { 1 } { 2 } x ^ { 4 } - x ^ { 3 } + x - 3$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) between \(x = 2\) and \(x = 2.5\)\\[0pt]
2. Starting with the interval [2,2.5] use interval bisection twice to find an interval of width 0.125 which contains \(\alpha\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\beta\) in the interval \([ - 2 , - 1 ]\).
3. Taking - 1.5 as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\beta\). Give your answer to 2 decimal places.\\ 4. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 2 x + 5 \right)$$
4. Find the four roots of \(\mathrm { f } ( x ) = 0\)
5. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.

4. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } 1 & 2 \\ 3 & - 1 \\ 4 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r r } 2 & - 1 & 4 \\ 1 & 3 & 1 \end{array} \right)$$

1. find \(\mathbf { A B }\).
2. Explain why \(\mathbf { A B } \neq \mathbf { B A }\).
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { c r } 2 k & - 2 \\ 3 & k \end{array} \right) \text {, where } k \text { is a real number }$$ find \(\mathbf { C } ^ { - 1 }\), giving your answer in terms of \(k\).

2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & - 4 \end{array} \right) , \mathbf { B } = \binom { 5 } { 3 }\) and \(\mathbf { C } = \left( \begin{array} { r r } 3 & 0 \\ - 2 & 2 \end{array} \right)\). Find

1. \(\mathbf { A B }\),
2. \(\mathbf { B A } - 4 \mathbf { C }\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 4 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & 0 \\ 3 & 2 \end{array} \right)\). Find

1. \(\mathbf { A B }\),
2. \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\).

2 You are given that \(\mathbf { A } = \left( \begin{array} { r } 4 \\ - 2 \\ 4 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 5 & 1 \\ 2 & - 3 \end{array} \right) , \mathbf { C } = \left( \begin{array} { l l l } 5 & 1 & 8 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { r r } - 2 & 0 \\ 4 & 1 \end{array} \right)\).

1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.

\(\mathbf { 1 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & p & - 4 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 0 & q \\ 2 & - 2 \\ 1 & - 3 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence prove that matrix multiplication is not commutative.

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 1 \\ 0 & 5 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 4 \\ 7 & 1 \end{array} \right]$$ \section*{Calculate AB} Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l l } 3 & 5 \\ 7 & 6 \end{array} \right] \quad \left[ \begin{array} { c c } 0 & 20 \\ 21 & 12 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 4 \\ 0 & 5 \end{array} \right] \quad \left[ \begin{array} { c c } 7 & 13 \\ 35 & 5 \end{array} \right]$$

1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1 \\ 7 & 2 \\ - 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2 \\ - 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1 \\ 4 & 3 & 8 \\ - 6 & 11 & 2 \end{array} \right)$$ Given that \(\mathbf { I }\) is the \(3 \times 3\) identity matrix,

1. 1. show that there is an integer \(k\) for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of \(k\)
   2. explain why there can be no constant \(m\) such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
2. 1. Show how the matrix \(\mathbf { C }\) can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14 \\ 4 x + 3 y + 8 z & = 3 \\ - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
   2. Hence use your calculator to solve these equations.

6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left[ \begin{array} { c c } 5 & 2 \\ - 3 & 4 \end{array} \right]$$ 6

1. \(\quad\) Find \(\operatorname { det } \mathbf { A }\) 6
2. Find \(\mathbf { A } ^ { - 1 }\) 6
3. Given that \(\mathbf { A B } = \left[ \begin{array} { c c } 9 & 6 \\ 5 & 12 \end{array} \right]\) and \(\mathbf { M } = 2 \mathbf { A } + \mathbf { B }\) find the matrix \(\mathbf { M }\)

11 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } 3 \mathrm { i } & - 2 \\ a & - \mathrm { i } \end{array} \right] \quad \text { and } \quad \mathbf { B } = \left[ \begin{array} { c c } 4 & 5 \\ - 2 \mathrm { i } & - 1 \end{array} \right]$$ where \(a\) is a real number. Calculate the product \(\mathbf { A B }\) in terms of \(a\) Give your answer in its simplest form.
[0pt] [3 marks]