Matrices

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]

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 matrix A is given by \(A = \begin{pmatrix} a & 1 \\ 1 & a \end{pmatrix}\), where \(a \neq \frac{1}{2}\), and I denotes the \(2 \times 2\) identity matrix. Find

1. \(2A - 3I\), [3]
2. \(A^{-1}\). [2]

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 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 \(q\).

5 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are non-singular square matrices, simplify $$\mathbf { A B } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) ^ { - 1 } .$$

Matrices A and B are given by \(\mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}\). Use A and B to disprove the proposition: "Matrix multiplication is commutative". [2]

6 The set \(S\) consists of all non-singular \(2 \times 2\) real matrices \(\mathbf { A }\) such that \(\mathbf { A Q } = \mathbf { Q A }\), where $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$$

1. Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\).
2. State clearly the restriction on the value of \(a\) such that \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\) is in \(S\).
3. Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)

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

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] $$\left[ \begin{array} { l l } 3 & 5 \\ 7 & 6 \end{array} \right] \quad \left[ \begin{array} { c c } 0 & 20 \\ 21 & 12 \end{array} \right] \quad \left[ \begin{array} { l l } 0 & 4 \\ 0 & 5 \end{array} \right] \quad \left[ \begin{array} { c c } 7 & 13 \\ 35 & 5 \end{array} \right]$$

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]

1 Given that \(\mathbf { M } \binom { x } { y } = \binom { 1 } { 3 }\), where \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 3 \\ 8 & 21 \end{array} \right)\), find \(x\) and \(y\).

1

1. Find the inverse of the matrix \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)\).
2. Use this inverse to solve the simultaneous equations $$\begin{aligned} 4 x + 3 y & = 5 \\ x + 2 y & = - 4 \end{aligned}$$ showing your working clearly.

7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$ Find the rank of \(\mathbf { M }\) and a basis for the null space of T . The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.

1. $$\mathbf { M } = \left( \begin{array} { c c } x & x - 2 \\ 3 x - 6 & 4 x - 11 \end{array} \right)$$ Given that the matrix \(\mathbf { M }\) is singular, find the possible values of \(x\).

1 Which of the following matrices is singular?
Circle your answer. \(\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l } 1 & 1 \\ 2 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c } 1 & - 2 \\ 1 & 2 \end{array} \right]\)

S is a singular matrix such that \(\det \mathbf{S} = \begin{vmatrix} a & a & x \\ x-b & a-b & x+1 \\ x^2 & a^2 & ax \end{vmatrix}\) Express the possible values of \(x\) in terms of \(a\) and \(b\). [7 marks]

1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3 \\ - 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?

1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3 \\ - 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?

3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4 \\ 1 & 1 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular. The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\).
   Given that the area of triangle \(R\) is 10 square units,
2. find the area of triangle \(S\). Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle \(R\) is transformed to the triangle \(T\) by the matrix \(\mathbf { B }\),
3. find, without evaluating \(\mathbf { B }\), the area of triangle \(T\).

Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]

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 \(( 0,0 )\).
   1. Find the value of \(p\).
   2. 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. $$\mathbf { M } = \left( \begin{array} { c c } 2 k + 1 & k \\ k + 7 & k + 4 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Show that \(\mathbf { M }\) is non-singular for all real values of \(k\).
2. Determine \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).

3 By using the determinant of an appropriate matrix, find the values of \(k\) for which the simultaneous equations $$\begin{aligned} & k x + 8 y = 1 \\ & 2 x + k y = 3 \end{aligned}$$ do not have a unique solution.

Which of the following matrices is an identity matrix? Circle your answer. [1 mark] \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)

\(A = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}\)

1. Find \(A^{-1}\) [2 marks]
2. The determinant of \(A^2\) is equal to 4. Find the possible values of \(k\). [3 marks]

11 Prove by induction that, for all integers \(n \geq 1\), $$\left( \mathbf { A B A } ^ { - 1 } \right) ^ { n } = \mathbf { A B } ^ { n } \mathbf { A } ^ { - 1 }$$ where \(\mathbf { A }\) and \(\mathbf { B }\) are square matrices of equal dimensions, and \(\mathbf { A }\) is non-singular.

$$\mathbf{M} = \begin{pmatrix} 3x & 7 \\ 4x + 1 & 2 - x \end{pmatrix}$$ Find the range of values of \(x\) for which the determinant of the matrix \(\mathbf{M}\) is positive. [5]

At the beginning of the year John had a total of £2000 in three different accounts. He has twice as much money in the current account as in the savings account. • The current account has an interest rate of 2.5% per annum. • The savings account has an interest rate of 3.7% per annum. • The supersaver account has an interest rate of 4.9% per annum. John has predicted that he will earn a total interest of £92 by the end of the year.

1. Model this situation as a matrix equation. [2]
2. Find the amount that John had in each account at the beginning of the year. [2]
3. In fact, the interest John will receive is £92 **to the nearest pound**. Explain how this affects the calculations. [2]

5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k \\ k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k \\ k & k \end{array} \right]$$ where \(k\) is a constant.

1. Find, in terms of \(k\) :
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A } ^ { 2 }\).
2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
3. It is now given that \(k = 1\).
   1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
   2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.

10

1. Show that \(\operatorname { det } \mathbf { A } = a + \mathrm { i }\) where \(a\) is an integer to be determined. 10 Matrix A is given by 10
2. Matrix B is given by $$\mathbf { B } = \left[ \begin{array} { c c } 14 - 2 \mathrm { i } & b \\ c & d \end{array} \right] \quad \text { and } \quad \mathbf { A B } = p$$ where \(b , c , d \in \mathbb { C }\) and \(p \in \mathbb { N }\) Find \(b , c , d\) and \(p\)

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

1. Find the value of a such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix \(\mathbf { B }\). Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
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