4.03b Matrix operations: addition, multiplication, scalar

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 matrix M is defined by \(\mathbf{M} = \begin{pmatrix} 3 & 2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) for all integers \(n \geq 1\) [5 marks]

The vector \(\mathbf{v}\) is an eigenvector of the matrix \(\mathbf{N}\) with corresponding eigenvalue 4 The vector \(\mathbf{v}\) is also an eigenvector of the matrix \(\mathbf{M}\) with corresponding eigenvalue 3 Given that $$\mathbf{N}\mathbf{M}^2\mathbf{v} = \lambda\mathbf{v}$$ find the value of \(\lambda\) Circle your answer. [1 mark] 10 \(\quad\) 24 \(\quad\) 36 \(\quad\) 144

The matrix M is given by $$\mathbf{M} = \frac{1}{10} \begin{pmatrix} a & a & -6 \\ 0 & 10 & 0 \\ 9 & 14 & -13 \end{pmatrix}$$ where \(a\) is a real number. The vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are eigenvectors of \(\mathbf{M}\) The corresponding eigenvalues are \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) respectively. It is given that \(\lambda_2 = 1\) and \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_3 = \begin{pmatrix} c \\ 0 \\ 1 \end{pmatrix}\), where \(c\) is an integer.

1. 1. Find the value of \(\lambda_1\) [2 marks]
   2. Find the value of \(a\) [2 marks]
2. Find the integer \(c\) and the value of \(\lambda_3\) [4 marks]
3. Find matrices \(\mathbf{U}\), \(\mathbf{D}\) and \(\mathbf{U}^{-1}\), such that \(\mathbf{D}\) is diagonal and \(\mathbf{M} = \mathbf{UDU}^{-1}\) [3 marks]

The line \(L_1\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$$ The transformation T is represented by the matrix $$\begin{pmatrix} 2 & 1 & 0 \\ 3 & 4 & 6 \\ -5 & 2 & -3 \end{pmatrix}$$ The transformation T transforms the line \(L_1\) to the line \(L_2\)

1. Show that the angle between \(L_1\) and \(L_2\) is 0.701 radians, correct to three decimal places. [4 marks]
2. Find the shortest distance between \(L_1\) and \(L_2\) Give your answer in an exact form. [6 marks]

Given that \(\mathbf{M} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}\), prove that \(\mathbf{M}^n = \begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}\) for all \(n \in \mathbb{N}\) [5 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]

Josh and Zoe are solving the following mathematics problem: The curve \(C_1\) has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\) Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\) Show that Zoe is correct. [5 marks]

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

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

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]

Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]

You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).

1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]

It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).

1. Explain geometrically why this claim is true. [2]
2. Verify algebraically that this claim is true. [3]

In this question you must show detailed reasoning. M is the matrix \(\begin{pmatrix} 1 & 6 \\ 0 & 2 \end{pmatrix}\). Prove that \(\mathbf{M}^n = \begin{pmatrix} 1 & 3(2^{n+1} - 2) \\ 0 & 2^n \end{pmatrix}\), for any positive integer \(n\). [6]

The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

1. Write down the matrix representing A. [1]
2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]

$$\text{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$$ [2] Prove by induction that \(\text{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]

$$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]

You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).

1. Find \(\mathbf{A}^4\). [1]
2. Describe the transformation that \(\mathbf{A}\) represents. [2]

The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).

1. Write down the matrix \(\mathbf{B}\). [1]

The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).

1. Find the coordinates of \(P'\). [1]

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

1. Find \(\mathbf{A} + 3\mathbf{B}\). [2]
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. [2]

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

1. Find \(\mathbf{A}^2\) and \(\mathbf{A}^3\). [3]
2. Hence suggest a suitable form for the matrix \(\mathbf{A}^n\). [1]
3. Use induction to prove that your answer to part (ii) is correct. [4]

The matrix \(\mathbf{M}\) is defined by \(\mathbf{M} = \begin{pmatrix} 3 & 2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) $$\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 3^n & 3^n - 1 & -3^n + 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) for all integers \(n \geq 1\) [5 marks]

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

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