4.03c Matrix multiplication: properties (associative, not commutative)

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

8 Three transformations, \(T _ { A } , T _ { B }\) and \(T _ { C }\), are represented by the matrices \(A , B\) and \(\mathbf { C }\) respectively. You are given that \(\mathbf { A } = \left( \begin{array} { l l } 1 & 0 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right)\).

1. Find the matrix which represents the inverse transformation of \(T _ { A }\).
2. By considering matrix multiplication, determine whether \(T _ { A }\) followed by \(T _ { B }\) is the same transformation as \(T _ { B }\) followed by \(T _ { A }\). Transformations R and S are each defined as being the result of successive transformations, as specified in the table.

   Transformation	First transformation	followed by
   R	\(\mathrm { T } _ { \mathrm { A } }\) followed by \(\mathrm { T } _ { \mathrm { B } }\)	\(\mathrm { T } _ { \mathrm { C } }\)
   S	\(\mathrm { T } _ { \mathrm { A } }\)	\(\mathrm { T } _ { \mathrm { B } }\) followed by \(\mathrm { T } _ { \mathrm { C } }\)

3. Explain, using a property of matrix multiplication, why R and S are the same transformations. A quadrilateral, \(Q\), has vertices \(D , E , F\) and \(G\) in anticlockwise order from \(D\). Under transformation \(\mathrm { R } , Q ^ { \prime }\) s image, \(Q ^ { \prime }\), has vertices \(D ^ { \prime } , E ^ { \prime } , F ^ { \prime }\) and \(G ^ { \prime }\) (where \(D ^ { \prime }\) is the image of \(D\), etc). The area of \(Q\), in suitable units, is 5 . You are given that det \(\mathbf { C } = a ^ { 2 } + 1\) where \(a\) is a real constant.
4. 1. Determine the order of the vertices of \(Q ^ { \prime }\), starting anticlockwise from \(D ^ { \prime }\).
   2. Find, in terms of \(a\), the area of \(Q ^ { \prime }\).
   3. Explain whether the inverse transformation for R exists. Justify your answer.

4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)

1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
3. 1. Find the exact area of the region \(R ^ { \prime }\).
   2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.

1 Two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), are given by \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & - 1 \\ 2 & - 3 & 1 \\ a & 1 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 6 & 3 & - 4 \\ - 1 & 6 & - 4 \\ 8 & - 8 & - 1 \end{array} \right)\) where \(a\) is a constant. Find the value of \(a\) for which \(\mathbf { A B } = \mathbf { B A }\).

6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Define the transformation represented by \(\mathbf { A }\).
2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.

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 \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).

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.

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.

3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1 \\ 2 & 0 \end{array} \right)\), where \(k\) is a constant.

1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.

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.

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0 & 2 \\ 3 & - 1 \end{array} \right)\).
Find

• the eigenvalues of \(\mathbf { A }\),
• an eigenvector associated with each eigenvalue.

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } 5 & - 8 \\ 2 & - 3 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 4 n + 1 & - 8 n \\ 2 n & 1 - 4 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21

1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. 1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
   2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
2. Determine \(\mathbf { R }\).
3. 1. Evaluate the determinant of \(\mathbf { R }\).
   2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)\).

1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).

7 The set \(M\) contains all matrices of the form \(\mathbf { X } ^ { n }\), where \(\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)\) and \(n\) is a positive integer.

1. Show that \(M\) contains exactly 12 elements.
2. Deduce that \(M\), together with the operation of matrix multiplication, form a cyclic group \(G\).
3. Determine all the proper subgroups of \(G\). \section*{END OF QUESTION PAPER}

4 M is the set of all \(2 \times 2\) matrices \(\mathrm { m } ( a , b )\) where \(a\) and \(b\) are rational numbers and $$\mathrm { m } ( a , b ) = \left( \begin{array} { l l } a & b \\ 0 & \frac { 1 } { a } \end{array} \right) , a \neq 0$$

1. Show that under matrix multiplication M is a group. You may assume associativity of matrix multiplication.
2. Determine whether the group is commutative. The set \(\mathrm { N } _ { k }\) consists of all \(2 \times 2\) matrices \(\mathrm { m } ( k , b )\) where \(k\) is a fixed positive integer and \(b\) can take any integer value.
3. Prove that \(\mathrm { N } _ { k }\) is closed under matrix multiplication if and only if \(k = 1\). Now consider the set P consisting of the matrices \(\mathrm { m } ( 1,0 ) , \mathrm { m } ( 1,1 ) , \mathrm { m } ( 1,2 )\) and \(\mathrm { m } ( 1,3 )\). The elements of P are combined using matrix multiplication but with arithmetic carried out modulo 4 .
4. Show that \(( \mathrm { m } ( 1,1 ) ) ^ { 2 } = \mathrm { m } ( 1,2 )\).
5. Construct the group combination table for P . The group R consists of the set \(\{ e , a , b , c \}\) combined under the operation *. The identity element is \(e\), and elements \(a , b\) and \(c\) are such that $$a ^ { * } a = b ^ { * } b = c ^ { * } c \quad \text { and } \quad a ^ { * } c = c ^ { * } a = b$$
6. Determine whether R is isomorphic to P . Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}

2 Which of the following statements is true about the operation of matrix multiplication on the set of all \(2 \times 2\) real matrices? Tick ( \(\checkmark\) ) one box. Matrix multiplication is associative and commutative. \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-03_109_112_552_1599} Matrix multiplication is associative but not commutative. □ Matrix multiplication is commutative but not associative. □ Matrix multiplication is not commutative and not associative. □

2

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).

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows. $$\mathbf{A} = \begin{bmatrix} p - 2 & p - 1 \\ 0 & 1 \end{bmatrix} \quad\quad \mathbf{B} = \begin{bmatrix} 1 & 2p - 1 \\ 0 & 4 - p \end{bmatrix}$$ Find the values of \(p\) such that \(\mathbf{A}\) and \(\mathbf{B}\) are commutative under matrix multiplication. Fully justify your answer. [4 marks]

Matrices A and B are given by \(\mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}\). Use A and B to disprove the proposition: "Matrix multiplication is commutative". [2]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find

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]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find
   [3]