Matrix arithmetic operations

2. (i) $$\mathbf { A } = \left( \begin{array} { c c } 2 k + 1 & k \\ - 3 & - 5 \end{array} \right) , \quad \text { where } k \text { is a constant }$$ Given that $$\mathbf { B } = \mathbf { A } + 3 \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, find

1. \(\mathbf { B }\) in terms of \(k\),
2. the value of \(k\) for which \(\mathbf { B }\) is singular.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r } 2
   - 3

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

1. Given that \(2 \mathbf { A } + \mathbf { B } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 2 \end{array} \right)\), write down the value of \(a\).
2. Given instead that \(\mathbf { A B } = \left( \begin{array} { l l } 7 & - 4 \\ 9 & - 7 \end{array} \right)\), find the value of \(a\). \end{itemize}

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

1. \(\mathbf { A } - 4 \mathbf { B }\),
2. BC ,
3. CA .

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

1. Find \(\mathbf { A } + 3 \mathbf { B }\).
2. Show that \(\mathbf { A } - \mathbf { B } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix and \(k\) is a constant whose value should be stated. \end{itemize}

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 4 & 1 \\ 5 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } - 3 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\).

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

1. Find \(3 \mathbf { A } - 4 \mathbf { B }\).
2. Find CB. Determine whether \(\mathbf { C B }\) is singular or non-singular, giving a reason for your answer.

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

1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
2. Show that matrix multiplication is not commutative.

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ a & 5 \end{array} \right)\). Find

1. \(\mathbf { A } ^ { - 1 }\),
2. \(2 \mathbf { A } - \left( \begin{array} { l l } 1 & 2 \\ 0 & 4 \end{array} \right)\).

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

1. \(2 \mathbf { A } + \mathbf { B }\),
2. \(\mathbf { A C }\),
3. CB. \end{itemize}

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 4 & 6 \\ 3 & - 5 \end{array} \right)\), and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Given that \(p \mathbf { A } + q \mathbf { B } = \mathbf { I }\), find the values of the constants \(p\) and \(q\).

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } a & 1 \\ 1 & 4 \end{array} \right)\), where \(a \neq \frac { 1 } { 4 }\), and \(\mathbf { I }\) denotes the \(2 \times 2\) identity matrix. Find

1. \(2 \mathbf { A } - 3 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 5 & 0 \\ 0 & 2 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find the values of the constants \(a\) and \(b\) for which \(a \mathbf { A } + b \mathbf { B } = \mathbf { I }\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a \\ 0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 2 & a \\ 4 & 1 \end{array} \right)\). I denotes the \(2 \times 2\) identity matrix. Find

1. \(\mathbf { A } + 3 \mathbf { B } - 4 \mathbf { I }\),
2. AB.

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } 2 & 1 \\ - 4 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. Find

1. \(4 \mathbf { A } - \mathbf { B } + 2 \mathbf { I }\),
2. \(\mathrm { A } ^ { - 1 }\),
3. \(\left( \mathbf { A B } ^ { - 1 } \right) ^ { - 1 }\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r } 6 \\ 4 \\ - 1 \end{array} \right)\). Find

1. \(5 \mathbf { A } - 3 \mathbf { B }\),
2. BC,
3. CA .

3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 4 & - 3 \\ - 2 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 3 & - 5 \\ 0 & 1 \end{array} \right)\).

1. Find 2A - 4B.
2. Write down the matrix \(\mathbf { C }\) such that \(\mathbf { A C } = 2 \mathbf { A }\).
3. Find the value of \(\operatorname { det } \mathbf { A }\).
4. In this question you must show detailed reasoning. Use \(\mathbf { A } ^ { - 1 }\) to solve the equations \(4 \mathrm { x } - 3 \mathrm { y } = 7\) and \(- 2 \mathrm { x } + 2 \mathrm { y } = 9\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } p & 2 \\ 4 & p \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right]$$

1. Find, in terms of \(p\), the matrices:
   1. \(\mathbf { A } - \mathbf { B }\);
   2. AB .
2. Show that there is a value of \(p\) for which \(\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, and state the corresponding value of \(k\).

1 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1

1. $$\left( \begin{array} { l l } x & 9 \\ y & z \end{array} \right) - 3 \left( \begin{array} { l l } z & y \\ z & y \end{array} \right) = k \mathbf { I }$$ where \(x , y , z\) and \(k\) are constants.
Determine the value of \(x\), the value of \(y\) and the value of \(z\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 4 \\ 4 & 3 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 2 \\ 2 & 0 \end{array} \right]$$

1. Calculate the matrices:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A B }\).
2. Show that \(\mathbf { A } + \mathbf { B } - \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   (2 marks)