4.03h Determinant 2x2: calculation

3 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathrm { A } | = | \mathrm { B } |\).
2. Verify that \(| \mathrm { AB } | = | \mathrm { A } | | \mathrm { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\). The \(2 \times 2\) matrix \(A\) represents a transformation \(T\) which has the following properties.
   The transformation \(S\) is represented by the matrix \(B\) where \(B = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).
   (b) Find the equation of the line of invariant points of S .
   (c) Show that any line of the form \(y = x + c\) is an invariant line of S .

2 In this question you must show detailed reasoning.
Solve the equation \(2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0\) giving each answer in the form \(\ln ( p + q \sqrt { r } )\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.

1. State the plane of reflection of \(R\).
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
   1. By using Euler's formula show that \(\cosh ( \mathrm { iz } ) = \cos z\).
   2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
   3. 1. Write down the general solution to (\textit{).
      2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where \(\lambda\) is a positive constant.
   4. In the case where \(\lambda = 16\) the door is held open at an angle of 0.9 radians and then released from rest at time \(t = 0\).
      1. Find, in a real form, the general solution of ( \(\dagger\) ).
      2. Find the particular solution of ( \(\dagger\) ).
      3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( \(\dagger\) ) improves the model.
      4. Find the value of \(\lambda\) for which the door is critically damped.

11

1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
   (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
   (b) Determine the value of \(n\).
   (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).

11

1. 1. Write down the matrix which represents a rotation through an angle \(\alpha\) anticlockwise about the origin.
   2. Show that the plane transformation given by the matrix $$\left( \begin{array} { c c } \cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\ \sin \theta - \cos \theta & \cos \theta + \sin \theta \end{array} \right)$$ is the composition of a rotation, \(R\), and a second transformation, \(S\). Describe both \(R\) and \(S\) fully.
2. 1. Write down the matrix which represents a reflection in the line \(y = x \tan \frac { 1 } { 2 } \beta\). For \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), the plane transformation \(T\) is given by the matrix $$\left( \begin{array} { c c } 1 + \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - 1 - \cos 2 \theta \end{array} \right)$$
   2. Show that \(T\) is the composition of a reflection and an enlargement, and describe these transformations in full.
   3. Find also the values of \(\theta\) for which \(T\) is an area-preserving transformation.

The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).

1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]

The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).

1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]

It is given that the eigenvalues of the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$ are \(1, 3, 4\). Find a set of corresponding eigenvectors. [4] Write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$ where \(n\) is a positive integer. [2] Find \(\mathbf{P}^{-1}\) and deduce that $$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]

The matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 10 & -7 & 10 \\ 7 & -5 & 8 \end{pmatrix},$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. [3] It is given that \(\begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\). Find the corresponding eigenvalue. [2] Find a diagonal matrix \(\mathbf{D}\) and matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) such that \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D}\). [5]

It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf{A}\) with \(\mathbf{e}\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf{B}\) for which \(\mathbf{e}\) is also a corresponding eigenvector.

1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf{A} + \mathbf{B}\) with \(\mathbf{e}\) as a corresponding eigenvector. [2]

The matrix \(\mathbf{A}\), given by $$\mathbf{A} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}$$ has \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) as eigenvectors.

1. Find the corresponding eigenvalues. [3]

The matrix \(\mathbf{B}\) has eigenvalues \(4\), \(5\) and \(1\) with corresponding eigenvectors \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) respectively.

1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} + \mathbf{B})^3 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\). [3]

It is given that $$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\). [1]
2. Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector. [3]
3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A} + \mathbf{A}^6\). [2]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).

1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
2. Find \(\mathbf{M}^T \mathbf{P}\). [3]

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

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}, \text{ where } k \text{ is constant.}$$ 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]

A point \(P\) is mapped onto a point \(Q\) under \(T\). The point \(Q\) has position vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) relative to an origin \(O\). Given that \(k = 3\),

1. find the position vector of \(P\). [3]

$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$

1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]

The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).

1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]

$$\mathbf{M} = \begin{pmatrix} 4 & -5 \\ 6 & -9 \end{pmatrix}$$

1. Find the eigenvalues of \(\mathbf{M}\). [4]

A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{M}\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).

1. Find a cartesian equation of this line. [3]

The transformation \(R\) is represented by the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.$$

1. Find the eigenvectors of \(\mathbf{A}\). [5]
2. Find an orthogonal matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}.$$ [5]
3. Hence describe the transformation \(R\) as a combination of geometrical transformations, stating clearly their order. [4]

Find the eigenvalues of the matrix \(\begin{pmatrix} 7 & 6 \\ 6 & 2 \end{pmatrix}\) (Total 4 marks)

Which of the following expressions is the determinant of the matrix \(\begin{bmatrix} a & 2 \\ b & 5 \end{bmatrix}\)? Circle your answer. [1 mark] \(5a - 2b\) \quad \(2a - 5b\) \quad \(5b - 2a\) \quad \(2b - 5a\)

Roberto is solving this mathematics problem: Roberto's solution is as follows:

1. Sketch the curve \(C_1\) [2 marks]
2. Explain what Roberto has done wrong. [2 marks]
3. Find the area enclosed by \(C_1\) [2 marks]
4. \(P\) and \(Q\) are distinct points on \(C_1\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\) [2 marks]
5. The matrix \(\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\) represents the transformation T T maps \(C_1\) onto a curve \(C_2\)
   1. T maps \(P\) onto the point \(P'\) Find the polar coordinates of \(P'\) [4 marks]
   2. Find the area enclosed by \(C_2\) Fully justify your answer. [2 marks]

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]

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]

A transformation T is represented by the matrix \(\mathbf{T}\) where \(\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q'\). Find the smallest possible value of the area of \(Q'\). [5]