4.03n Inverse 2x2 matrix

The matrix M is defined as $$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ -1 & -1 & -2 \\ 1 & 2 & c \end{pmatrix}$$ where \(c\) is a real number.

1. The linear transformation T is represented by the matrix \(\mathbf{M}\) Show that, for one particular value of \(c\), the image under T of every point lies in the plane $$x + 5y + 3z = 0$$ State the value of \(c\) for which this occurs. [3 marks]
2. It is given that M is a non-singular matrix.
   1. State any restrictions on the value of \(c\) [2 marks]
   2. Find \(\mathbf{M}^{-1}\) in terms of \(c\) [4 marks]
   3. Using your answer from part (b)(ii), solve $$2x - y + z = -3$$ $$-x - y - 2z = -6$$ $$x + 2y + 4z = 13$$ [3 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]

\(\mathbf{A}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\)

1. Using the result $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$ show that $$(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}$$ [3 marks]
2. It is given that \(\mathbf{A} = \begin{pmatrix} 4 & 5 \\ -1 & k \end{pmatrix}\), where \(k\) is a real constant.
   1. Find \((\mathbf{A}^{-1})^T\), giving your answer in terms of \(k\) [2 marks]
   2. State the restriction on the possible values of \(k\) [1 mark]

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 matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 3 & 3 & 0 \\ 0 & 2 & 2 \\ 1 & 3 & 4 \end{pmatrix}\).

1. Determine the characteristic equation of \(\mathbf{A}\). [3]
2. Hence verify that the eigenvalues of \(\mathbf{A}\) are 1, 2 and 6. [1]
3. For each eigenvalue of \(\mathbf{A}\) determine an associated eigenvector. [4]
4. Use the results of parts (b) and (c) to find \(\mathbf{A}^n\) as a single matrix, where \(n\) is a positive integer. [6]

In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.

1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
2. Find a vector equation for the axis of the rotation. [4]
3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]

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]

$$\mathbf{M} = \begin{pmatrix} -2 & 5 \\ 6 & k \end{pmatrix}$$ where \(k\) is a constant. Given that $$\mathbf{M}^2 + 11\mathbf{M} = a\mathbf{I}$$ where \(a\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,

1. 1. determine the value of \(a\)
   2. show that \(k = -9\) [3]
2. Determine the equations of the invariant lines of the transformation represented by \(\mathbf{M}\). [6]
3. State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer. [1]

$$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}$$

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

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) is constant. A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).

1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [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]

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

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

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.

1. Write down A. [1]
2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]

Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.

1. Describe T. [1]
2. Determine the matrix which represents the transformation S followed by T. [2]
3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]

A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.

1. Explain why the suggestion by Student P might not work. [2]

Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.

1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]

Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).

1. 1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
   2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
   3. Find a different function \(f\) that will satisfy the requirements of the task. [4]

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the \(x\)-axis. A is the matrix which represents S.

1. Write down A. [1]
2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]

Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.

1. Describe T. [1]
2. Determine the matrix which represents the transformation S followed by T. [2]
3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]