Matrix satisfying given equation

7. (i) $$\mathbf { A } = \left( \begin{array} { r r } 6 & k \\ - 3 & - 4 \end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$ Find, in terms of \(k\),

1. \(\mathbf { A } ^ { - 1 }\)
2. \(\mathbf { A } ^ { 2 }\) Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9 \\ - 3 & - 5 \end{array} \right)\)
3. find the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } - \frac { 1 } { 2 } & - \sqrt { 3 } \\ \frac { \sqrt { 3 } } { 2 } & - 1 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\).
4. Find the value of \(p\).
5. Find the value of \(\theta\).

3. $$\mathbf { A } = \left( \begin{array} { l l } 4 & - 2 \\ a & - 3 \end{array} \right)$$ where \(a\) is a real constant and \(a \neq 6\)

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { A } + 2 \mathbf { A } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

1. $$\mathbf { A } = \left( \begin{array} { r r } 3 & a \\ - 2 & - 2 \end{array} \right)$$ where \(a\) is a non-zero constant and \(a \neq 3\)

1. Determine \(\mathbf { A } ^ { - 1 }\) giving your answer in terms of \(a\). Given that \(\mathbf { A } + \mathbf { A } ^ { - 1 } = \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. determine the value of \(a\).

7. Given that \(\mathbf { X } = \left( \begin{array} { r r } 2 & a \\ - 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),

1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

7. Given that \(\mathbf { X } = \left( \begin{array} { c c } 2 & a \\ - 1 & - 1 \end{array} \right)\), where \(a\) is a constant, and \(a \neq 2\),

1. find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { X } + \mathbf { X } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

8. $$\mathbf { A } = \left( \begin{array} { c c } 6 & - 2 \\ - 4 & 1 \end{array} \right)$$ and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Prove that $$\mathbf { A } ^ { 2 } = 7 \mathbf { A } + 2 \mathbf { I }$$
2. Hence show that $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 2 } ( \mathbf { A } - 7 \mathbf { I } )$$ The transformation represented by \(\mathbf { A }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( 2 k + 8 , - 2 k - 5 )\), where \(k\) is a constant,
3. find, in terms of \(k\), the coordinates of \(P\).

5. (i) $$\mathbf { A } = \left( \begin{array} { l l } p & 2 \\ 3 & p \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 5 & 4 \\ 6 & - 5 \end{array} \right)$$ where \(p\) is a constant.

1. Find, in terms of \(p\), the matrix \(\mathbf { A B }\) Given that $$\mathbf { A B } + 2 \mathbf { A } = k \mathbf { I }$$ where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(p\) and the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { r r } a & - 9 \\ 1 & 2 \end{array} \right) , \text { where } a \text { is a real constant }$$ Triangle \(T\) has an area of 15 square units.
   Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of triangle \(T ^ { \prime }\) is 270 square units, find the possible values of \(a\).

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

1. Find \(\mathbf { A } ^ { 2 }\) and verify that \(\mathbf { A } ^ { 2 } = 4 \mathbf { A } - \mathbf { I }\).
2. Hence, or otherwise, show that \(\mathbf { A } ^ { - 1 } = 4 \mathbf { I } - \mathbf { A }\).

4 It is given that $$\mathbf { A } = \left[ \begin{array} { l l } 1 & 4 \\ 3 & 1 \end{array} \right]$$ and that \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Show that \(( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }\) for some integer \(k\).
2. Given further that $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 3 \\ p & 1 \end{array} \right]$$ find the integer \(p\) such that $$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$

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

3

1. You are given two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2 \\ 2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5 \\ 1 & 1 & 3 \\ - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2 \\ - 5 & 9 & - 1 \\ 3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).