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. (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 \(\mathbf { A }\) 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]

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

1. \(\mathbf{AB}\), [2]
2. \(\mathbf{BA} - 4\mathbf{C}\). [4]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$

1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows: $$\mathbf{A} = \begin{bmatrix} x + 1 & 2 \\ x + 2 & -3 \end{bmatrix}$$ $$\mathbf{B} = \begin{bmatrix} x - 4 & x - 2 \\ 0 & -2 \end{bmatrix}$$ Show that there is a value of \(x\) for which \(\mathbf{AB} = k\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix and \(k\) is an integer to be found. [3 marks]

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 & 3 \end{pmatrix}.$$ Calculate all possible products formed from two of these three matrices. [4]

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = (1 \quad 3).$$ Calculate all possible products formed from two of these three matrices. [4]