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

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = ( - 9 \mathbf { i } + 10 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \\ l _ { 2 } : & \mathbf { r } = ( 3 \mathbf { i } + \mathbf { j } + 17 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{array}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\) has position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\). The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\).
4. Find the position vector of \(B\).

7. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0 \\ 9 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\).
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

6.

1. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
   1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
   2. Describe fully the single transformation represented by the matrix \(\mathbf { B }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
   3. Find \(\mathbf { C }\).
   4. \(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 \\ 1 & k \end{array} \right)\), where \(k\) is a real number. Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).

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

1. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
2. Hence find the smallest positive integer value of \(n\) for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\). Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4 \\ - 3 & - 5 \end{array} \right)\),
3. find the matrix \(\mathbf { B }\).

7. The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 5 \\ - 3 & 2 \end{array} \right)$$ The transformation represented by \(\mathbf { A }\) maps triangle \(T\) onto triangle \(T ^ { \prime }\) Given that the area of triangle \(T\) is \(23 \mathrm {~cm} ^ { 2 }\)

1. determine the area of triangle \(T ^ { \prime }\) (2) The point \(P\) has coordinates ( \(3 p + 2,2 p - 1\) ) where \(p\) is a constant. The transformation represented by \(\mathbf { A }\) maps \(P\) onto the point \(P ^ { \prime }\) with coordinates \(( 17 , - 18 )\)
2. Determine the value of \(p\). Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$
3. describe fully the single geometrical transformation represented by matrix \(\mathbf { B }\) The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { C }\) is equivalent to the transformation represented by matrix \(\mathbf { B }\)
4. Determine C \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-21_2255_50_314_34}
   

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

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + n - 1 \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\left( \begin{array} { c c } 7 & - 12 \\ 3 & - 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 6 n + 1 & - 12 n \\ 3 n & 1 - 6 n \end{array} \right)$$

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\). \section*{II}

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

$$\left( \begin{array} { l l } 1 & r \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & \left( 2 ^ { n } - 1 \right) r \\ 0 & 2 ^ { n } \end{array} \right)$$ where \(r\) is a constant. $$\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 5 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 2 \end{array} \right) ^ { 4 }$$ The transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\) is represented by the matrix \(\mathbf { B }\) (b) (i) Determine \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\) (c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
   
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6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
2. show that \(k = 3\),
3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
   The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).

8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)\).

1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.

1. Describe fully the transformations P and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
   Triangle T is transformed first by transformation P , then by transformation Q .
2. Find the single matrix that represents this composite transformation.
3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.

1

1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix for a reflection in the line \(y = x\).
3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
4. What single transformation is equivalent to this composite transformation?

1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation B .

1. Describe the transformations A and B .
2. Find the matrix representing the composite transformation consisting of A followed by B .
3. What single transformation is represented by this matrix?

3 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } \lambda & 6 & - 4 \\ 2 & 5 & - 1 \\ - 1 & 4 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } - 19 & 34 & - 14 \\ 5 & - 5 & 5 \\ - 13 & 18 & - 3 \end{array} \right)\) and \(\mathbf { A B } = \mu \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity
matrix.

1. Find the values of \(\lambda\) and \(\mu\).
2. Hence find \(\mathbf { B } ^ { - 1 }\).

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 1 \\ 0 & - 1 & 2 \\ 0 & 0 & 1 \end{array} \right)$$ and find corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
[0pt] [Question 11 is printed on the next page.]

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 3 & - 4 \\ 2 & - 4 & 7 & - 9 \\ 4 & - 8 & 14 & - 18 \\ 5 & - 10 & 17 & - 22 \end{array} \right)$$ Find the rank of \(\mathbf { M }\). Obtain a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right)$$ and hence show that any solution of $$\mathbf { M x } = \left( \begin{array} { l } 15 \\ 33 \\ 66 \\ 81 \end{array} \right)$$

5 The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & - 2 \\ 6 & 4 & - 6 \\ 6 & 5 & - 7 \end{array} \right)$$ has eigenvalues \(1 , - 1\) and - 2 .

1. Find a set of corresponding eigenvectors.
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \mathbf { A } - 2 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Write down the eigenvalues of \(\mathbf { B }\), and state a set of corresponding eigenvectors.

One of the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 4 & 6 \\ 2 & - 4 & 2 \\ - 3 & 4 & a \end{array} \right)$$ is - 2 . Find the value of \(a\). Another eigenvalue of \(\mathbf { A }\) is - 5 . Find eigenvectors \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) corresponding to the eigenvalues - 2 and - 5 respectively. The linear space spanned by \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) is denoted by \(V\).

1. Prove that, for any vector \(\mathbf { x }\) belonging to \(V\), the vector \(\mathbf { A x }\) also belongs to \(V\).
2. Find a non-zero vector which is perpendicular to every vector in \(V\), and determine whether it is an eigenvector of \(\mathbf { A }\).

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 2 & - 1 & \alpha \\ 2 & 3 & - 1 & 0 \\ 2 & 1 & 2 & - 2 \\ 0 & 1 & - 3 & - 2 \end{array} \right)$$ Given that the dimension of the range space of T is 4 , show that \(\alpha \neq 1\). It is now given that \(\alpha = 1\). Show that the vectors $$\left( \begin{array} { l } 1 \\ 2 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 3 \\ 1 \\ 1 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 2 \\ - 3 \end{array} \right)$$ form a basis for the range space of T . Given also that the vector \(\left( \begin{array} { c } p \\ 1 \\ 1 \\ q \end{array} \right)\) is in the range space of T , find a condition satisfied by \(p\) and \(q\).

9 Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 3 \end{array} \right)$$ Find a non-singular matrix \(\mathbf { M }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 2 \mathbf { I } ) ^ { 3 } = \mathbf { M D M } ^ { - 1 }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix.

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 3 & 4 \\ 1 & - 1 & 2 & 3 \\ 1 & - 3 & 3 & 5 \\ 1 & 4 & 2 & 2 \end{array} \right)$$ The range space of T is denoted by \(V\).

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \\ 4 \end{array} \right) , \left( \begin{array} { l } 3 \\ 2 \\ 3 \\ 2 \end{array} \right)\) are a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
3. State, with a reason, whether \(W\) is a vector space.
4. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).