4.03b Matrix operations: addition, multiplication, scalar

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1. Show that there is a value of \(t\) for which \(\mathbf { A B }\) is an integer multiple of the \(3 \times 3\) identity matrix \(\mathbf { I }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 2 & 1 \\ t & 1 & - t \\ 3 & 2 & 1 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { c r r } t - 2 & 0 & 5 \\ 12 & - 2 & - 6 \\ 3 t & 4 & 7 \end{array} \right) .$$
2. Express the system of equations $$\begin{aligned} - 5 x + 5 z & = 8 \\ 12 x - 2 y - 6 z & = 12 \\ - 9 x + 4 y + 7 z & = 22 \end{aligned}$$ in the form \(\mathbf { C x } = \mathbf { u }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, and \(\mathbf { x }\) and \(\mathbf { u }\) are suitable column vectors.
3. Use the result of part (i) to solve the system of equations given in part (ii).

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where \(p\) is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.

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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 \(H\).

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .

8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.

1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.

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]

$$\mathbf{P} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$$ The matrix \(\mathbf{P}\) represents the transformation \(U\)

1. Give a full description of \(U\) as a single geometrical transformation. [2]

The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf{Q}\), is a reflection in the line \(y = -x\)

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

The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf{R}\)

1. Determine the matrix \(\mathbf{R}\) [2]

The transformation \(W\) is represented by the matrix \(3\mathbf{R}\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T'\) The transformation \(W'\) maps the triangle \(T'\) back to the original triangle \(T\)

1. Determine the matrix that represents \(W'\) [3]

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Find \(\mathbf{D}\). [2]
2. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]

$$\mathbf{A} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \quad \mathbf{B} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$$ The transformation represented by \(\mathbf{B}\) followed by the transformation represented by \(\mathbf{A}\) is equivalent to the transformation represented by \(\mathbf{P}\).

1. Find the matrix \(\mathbf{P}\). [2]

Triangle \(T\) is transformed to the triangle \(T'\) by the transformation represented by \(\mathbf{P}\). Given that the area of triangle \(T'\) is 24 square units,

1. find the area of triangle \(T\). [3]

Triangle \(T'\) is transformed to the original triangle \(T\) by the matrix represented by \(\mathbf{Q}\).

1. Find the matrix \(\mathbf{Q}\). [2]

For \(n \in \mathbb{Z}^+\) prove that

1. \(2^{3n + 2} + 5^{n + 1}\) is divisible by 3, [9]
2. \(\begin{pmatrix} -2 & -1 \\ 9 & 4 \end{pmatrix}^n = \begin{pmatrix} 1-3n & -n \\ 9n & 3n+1 \end{pmatrix}\). [7]

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

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

1. 1. Find the matrix \(\mathbf{A}^2\). [1 mark]
   2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]

The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.

1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]

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]

\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).

1. Write down the matrix, X, for this transformation. [2]
2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
3. Find the matrix that represents transformation Q followed by transformation P. [2]

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

1. \(\mathbf{AB}\), [2]
2. \(\mathbf{BA} - 4\mathbf{C}\). [4]

You are given that \(\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}\) and that \(\mathbf{AB}\) is of the form \(\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}\).

1. Find the value of \(n\). [2]
2. Write down the inverse matrix \(\mathbf{A}^{-1}\) and state the condition on \(k\) for this inverse to exist. [4]
3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
   2x + y + 2z &= 12
   3x + 2y - z &= 3 \end{align} [5]

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]

Three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by $$\mathbf{A} = \begin{pmatrix} 5 & 2 & -3 \\ 0 & 7 & 6 \\ 4 & 1 & 0 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 3 & -5 \\ -2 & 6 \end{pmatrix} \quad \text{and } \mathbf{C} = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 2 & 0 \end{pmatrix}$$ Which of the following **cannot** be calculated? Circle your answer. [1 mark] \(\mathbf{AB}\) \(\qquad\) \(\mathbf{AC}\) \(\qquad\) \(\mathbf{BC}\) \(\qquad\) \(\mathbf{A}^2\)

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]