Solving matrix equations for unknown matrix

4. $$\mathbf { A } = \left( \begin{array} { c c } 2 p & 3 q \\ 3 p & 5 q \end{array} \right)$$ where \(p\) and \(q\) are non-zero real constants.

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\) and \(q\). Given \(\mathbf { X A } = \mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { c c } p & q \\ 6 p & 11 q \\ 5 p & 8 q \end{array} \right)$$
2. find the matrix \(\mathbf { X }\), giving your answer in its simplest form.

8. $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 2 & 3 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular.
2. Find \(\mathbf { B }\) such that \(\mathbf { B A } ^ { 2 } = \mathbf { A }\).

7. $$\mathbf { P } = \left( \begin{array} { c c } 3 a & - 2 a \\ - b & 2 b \end{array} \right) , \quad \mathbf { M } = \left( \begin{array} { c c } - 6 a & 7 a \\ 2 b & - b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.

1. Find \(\mathbf { P } ^ { - 1 }\), leaving your answer in terms of \(a\) and \(b\). Given that $$\mathbf { M } = \mathbf { P Q }$$
2. find the matrix \(\mathbf { Q }\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-19_95_77_2617_1804}

6. $$\mathbf { A } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } - 1 & 1 \\ 0 & 1 \end{array} \right)$$ Given that \(\mathbf { M } = ( \mathbf { A } + \mathbf { B } ) ( 2 \mathbf { A } - \mathbf { B } )\),

1. calculate the matrix \(\mathbf { M }\),
2. find the matrix \(\mathbf { C }\) such that \(\mathbf { M C } = \mathbf { A }\).

2. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 1 \\ 4 & 3 \end{array} \right) , \quad \mathbf { P } = \left( \begin{array} { r r } 3 & 6 \\ 11 & - 8 \end{array} \right)$$

1. Find \(\mathbf { A } ^ { - 1 }\) (2) The transformation represented by the matrix \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\) is equivalent to the transformation represented by the matrix \(\mathbf { P }\).
2. Find \(\mathbf { B }\), giving your answer in its simplest form.

3. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } - 2 & 3 \\ 1 & 1 \end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r } - 1 & 5 & 12 \\ 3 & - 5 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { - 1 }\)
2. Hence, or otherwise, find the matrix \(\mathbf { B }\), giving your answer in its simplest form.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
   1. describe fully the single geometrical transformation represented by the matrix \(\mathbf { C }\).
   2. Hence find the matrix \(\mathbf { C } ^ { 39 }\)

5. Given that \(a\) and \(b\) are non-zero constants and that $$\mathbf { X } = \left( \begin{array} { r r } a & 2 b \\ - a & 3 b \end{array} \right) ,$$

1. find \(\mathbf { X } ^ { - 1 }\), giving your answer in terms of \(a\) and \(b\). Given also that \(\mathbf { Z X } = \mathbf { Y }\), where \(\mathbf { Y } = \left( \begin{array} { c c } 3 a & b \\ a & 2 b \end{array} \right)\),
2. find \(\mathbf { Z }\), simplifying your answer.

6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 8 \end{array} \right)\).

1. Find \(\mathbf { C } ^ { - 1 }\).
2. Given that \(\mathbf { C } = \mathbf { A B }\), where \(\mathbf { A } = \left( \begin{array} { l l } 2 & 1 \\ 1 & 3 \end{array} \right)\), find \(\mathbf { B } ^ { - 1 }\).

4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 1 \\ 3 & 5 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { - 1 }\). The matrix \(\mathbf { B } ^ { - 1 }\) is given by \(\mathbf { B } ^ { - 1 } = \left( \begin{array} { r r } 1 & 1 \\ 4 & - 1 \end{array} \right)\).
2. Find \(( \mathbf { A B } ) ^ { - 1 }\).

3 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & a \\ 0 & 1 \end{array} \right)\), where \(a\) is a constant.

1. Find \(\mathbf { A } ^ { - 1 }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { l l } 2 & a \\ 4 & 1 \end{array} \right)\).
2. Given that \(\mathbf { P A } = \mathbf { B }\), find the matrix \(\mathbf { P }\).

3 In this question you must show detailed reasoning. \(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1 \\ 0 & 1 \end{array} \right)\), find \(\mathbf { B }\).

2

1. The matrices \(\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)\) are such that \(\mathbf { M } \mathbf { N } = \mathbf { I }\).
   Find \(a , b\) and \(c\).
2. State with a reason whether or not \(\mathbf { N }\) is the inverse of \(\mathbf { M }\).

1. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 3 & 7 \\ - 2 & 0 \end{array} \right)\), \(\mathbf { B } = \left( \begin{array} { l l } 5 & 1 \\ 0 & 4 \end{array} \right)\).

The matrix \(\mathbf { X }\) is such that \(\mathbf { A X } = \mathbf { B }\). Showing all your working, find the matrix \(\mathbf { X }\).

2. (a) The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left( \begin{array} { c c } 3 & 4 \\ - 1 & - 2 \end{array} \right) , \quad \mathbf { B } = \binom { - 11 } { 7 }$$ Given that \(\mathbf { A X } = \mathbf { B }\), find the matrix \(\mathbf { X }\).
(b) (i) Find the \(2 \times 2\) matrix, \(\mathbf { T }\), which represents a reflection in the line \(y = - 2 x\).
(ii) The images of the points \(C ( 2,7 )\) and \(D ( 3,1 )\), under \(\mathbf { T }\), are \(E\) and \(F\) respectively. Find the coordinates of the midpoint of \(E F\).

2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 2 & - 1 \\ 4 & - 7 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { c c c } 2 & 0 & 9 \\ 4 & - 20 & 13 \end{array} \right]\).

1. Find the inverse of \(\mathbf { A }\).
2. Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \mathbf { B }\).

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that \(\mathbf{A} = \begin{bmatrix} 4 & 2 \\ -1 & -3 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 4 & 2 \\ 2 & 1 \end{bmatrix}\).

1. Explain why \(\mathbf{B}\) has no inverse. [1]
2. 1. Find the inverse of \(\mathbf{A}\). [3]
   2. Hence, find the matrix \(\mathbf{X}\), where \(\mathbf{AX} = \begin{bmatrix} -4 \\ 1 \end{bmatrix}\) [2]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & a \\ 0 & 1 \end{pmatrix}\), where \(a\) is a constant.

1. Find \(\mathbf{A}^{-1}\). [2]

The matrix \(\mathbf{B}\) is given by \(\mathbf{B} = \begin{pmatrix} 2 & a \\ 4 & 1 \end{pmatrix}\).

1. Given that \(\mathbf{PA} = \mathbf{B}\), find the matrix \(\mathbf{P}\). [3]

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = I + 2\mathbf{A}$$ where \(I\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]