4.03b Matrix operations: addition, multiplication, scalar

7

1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$$
   1. Describe the transformation T geometrically.
   2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
   3. Explain briefly why the transformation T followed by T is the identity transformation.
2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$$
   1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
   2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$

1. Calculate:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { B A }\).
2. Describe fully the geometrical transformation represented by each of the following matrices:
   1. \(\mathbf { A }\);
   2. \(\mathbf { B }\);
   3. \(\mathbf { B A }\).

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.

3

1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
   1. a rotation about the origin through \(90 ^ { \circ }\) clockwise;
   2. a rotation about the origin through \(180 ^ { \circ }\).
2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { r r } 2 & 4 \\ - 1 & - 3 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } - 2 & 1 \\ - 4 & 3 \end{array} \right]$$
   1. Calculate the matrix \(\mathbf { A B }\).
   2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix, for some integer \(k\).
3. Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(( \mathbf { A } + \mathbf { B } ) ^ { 2 }\);
   3. \(( \mathbf { A } + \mathbf { B } ) ^ { 4 }\).

8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).

1. On the diagram, draw:
   1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
   2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right]$$
2. Find the matrix of:
   1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
   2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}

6

1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2 \\ 3 & 0 \end{array} \right]\).
   1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2 \\ 3 & 6 \end{array} \right]\), find the value of \(m\).
   2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right]\).
   1. Describe the geometrical transformation represented by \(\mathbf { A }\).
   2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1 \\ 1 & 1 \end{array} \right]\), where \(k\) is a surd.
   3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\). \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 1 \\ 3 & 8 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right]$$ The matrix \(\mathbf { M } = \mathbf { A } - 2 \mathbf { B }\).

1. Show that \(\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]\), where \(n\) is a positive integer.
   (2 marks)
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement of scale factor \(p\) and a reflection in a line \(L\). State the value of \(p\) and write down the equation of \(L\).
3. Show that $$\mathbf { M } ^ { 2 } = q \mathbf { I }$$ where \(q\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 0 & 2 \\ 2 & 0 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right]$$

1. Calculate the matrix \(\mathbf { A B }\).
2. Show that \(\mathbf { A } ^ { 2 }\) is of the form \(k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
3. Show that \(( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }\).

7

1. Using surd forms where appropriate, find the matrix which represents:
   1. a rotation about the origin through \(30 ^ { \circ }\) anticlockwise;
   2. a reflection in the line \(y = \frac { 1 } { \sqrt { 3 } } x\).
2. The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { c c } 1 & \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$ 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.
3. The transformation represented by \(\mathbf { A }\) is followed by the transformation represented by \(\mathbf { B }\), where $$\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 3 } & - 1 \\ 1 & \sqrt { 3 } \end{array} \right]$$ Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.

7 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left[ \begin{array} { c c } - 1 & - \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$

1. 1. Calculate the matrix \(\mathbf { A } ^ { 2 }\).
   2. Show that \(\mathbf { A } ^ { 3 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. Describe the single geometrical transformation, or combination of two geometrical transformations, corresponding to each of the matrices:
   1. \(\mathrm { A } ^ { 3 }\);
   2. A.

6

1. Using surd forms, find the matrix of a rotation about the origin through \(135 ^ { \circ }\) anticlockwise.
2. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { r r } - 1 & - 1 \\ 1 & - 1 \end{array} \right]\).
   1. Given that \(\mathbf { M }\) represents an enlargement followed by a rotation, find the scale factor of the enlargement and the angle of the rotation.
   2. The matrix \(\mathbf { M } ^ { 2 }\) also represents an enlargement followed by a rotation. State the scale factor of the enlargement and the angle of the rotation.
   3. Show that \(\mathbf { M } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   4. Deduce that \(\mathbf { M } ^ { 2012 } = - 2 ^ { n } \mathbf { I }\) for some positive integer \(n\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } p & 2 \\ 4 & p \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 3 & 1 \\ 2 & 3 \end{array} \right]$$

1. Find, in terms of \(p\), the matrices:
   1. \(\mathbf { A } - \mathbf { B }\);
   2. AB .
2. Show that there is a value of \(p\) for which \(\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, and state the corresponding value of \(k\).

5

1. The matrix \(\mathbf { A }\) is defined by \(\mathbf { A } = \left[ \begin{array} { c c } - 2 & c \\ d & 3 \end{array} \right]\).
   Given that the image of the point \(( 5,2 )\) under the transformation represented by \(\mathbf { A }\) is \(( - 2,1 )\), find the value of \(c\) and the value of \(d\).
   [0pt] [4 marks]
2. The matrix \(\mathbf { B }\) is defined by \(\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 } \\ - \sqrt { 2 } & \sqrt { 2 } \end{array} \right]\).
   1. Show that \(\mathbf { B } ^ { 4 } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   2. Describe the transformation represented by the matrix \(\mathbf { B }\) as a combination of two geometrical transformations.
   3. Find the matrix \(\mathbf { B } ^ { 17 }\). \(6 \quad \mathrm {~A}\) curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
3. Prove your conjecture.

2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & a \\ - 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 2 & 1 \end{array} \right)\), where \(a\) is
a constant. a constant.

1. By multiplying out the matrices on both sides of the equation, verify that \(\mathbf { A } ( \mathbf { B C } ) = ( \mathbf { A B } ) \mathbf { C }\).
2. State the property of matrix multiplication illustrated by this result.

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2 \\ 2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1 \\ 1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.

1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
   1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
   2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
   3. Hence verify your answer to part (i).

8

1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
   1. Find \(\mathbf { M } ^ { 2 }\).
   2. Write down the transformation represented by \(\mathbf { M }\).
   3. Hence state the geometrical significance of the result of part (i).
2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.

6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.

1. 1. Show that T reverses orientation.
   2. State, in terms of \(\lambda\), the area scale factor of T .
2. 1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
   2. Hence specify the transformation equivalent to two applications of T .
3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
   1. Determine the matrix associated with S .
   2. Hence describe the transformation S .

6 Prove by mathematical induction that \(\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)\) for all positive integers \(n\). Answer all the questions.
Section B (107 marks)

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.

1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
2. Specify the transformation of the plane associated with each of the following matrices.
   1. M
   2. N
3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.

2

1. The matrices \(\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)\) are such that \(\mathbf { M } \mathbf { N } = \mathbf { I }\).
   Find \(a , b\) and \(c\).
2. State with a reason whether or not \(\mathbf { N }\) is the inverse of \(\mathbf { M }\).

7. Using mathematical induction, prove that $$\left[ \begin{array} { l l } 2 & 5 \\ 0 & 2 \end{array} \right] ^ { n } = \left[ \begin{array} { c c } 2 ^ { n } & 2 ^ { n - 1 } \times 5 n \\ 0 & 2 ^ { n } \end{array} \right]$$ for all positive integers \(n\).

4. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 4 \\ k & 2 & - 2 \\ 4 & 1 & - 2 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r r } k - 7 & 6 & - 10 \\ 2 & - 20 & 24 \\ - 3 & 2 & - 1 \end{array} \right)$$ where \(k\) is a constant.

1. Determine, in simplest form in terms of \(k\), the matrix \(\mathbf { M N }\).
2. Given that \(k = 5\)
   1. write down \(\mathbf { M N }\)
   2. hence write down \(\mathbf { M } ^ { - 1 }\)
3. Solve the simultaneous equations $$\begin{aligned} & 2 x + y + 4 z = 2 \\ & 5 x + 2 y - 2 z = 3 \\ & 4 x + y - 2 z = - 1 \end{aligned}$$
4. Interpret the answer to part (c) geometrically.

1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1 \\ 7 & 2 \\ - 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2 \\ - 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1 \\ 4 & 3 & 8 \\ - 6 & 11 & 2 \end{array} \right)$$ Given that \(\mathbf { I }\) is the \(3 \times 3\) identity matrix,

1. 1. show that there is an integer \(k\) for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of \(k\)
   2. explain why there can be no constant \(m\) such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
2. 1. Show how the matrix \(\mathbf { C }\) can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14 \\ 4 x + 3 y + 8 z & = 3 \\ - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
   2. Hence use your calculator to solve these equations.

3. $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Describe fully the single geometric transformation \(A\) represented by the matrix \(\mathbf { A }\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$ The transformation \(B\) is represented by the matrix \(\mathbf { B }\).
   The transformation \(A\) followed by the transformation \(B\) is the transformation \(C\), which is represented by the matrix \(\mathbf { C }\). To determine matrix \(\mathbf { C }\), a student attempts the following matrix multiplication. $$\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right) \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$
2. State the error made by the student.
3. Determine the correct matrix \(\mathbf { C }\).