4.03b Matrix operations: addition, multiplication, scalar

4 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a rotation of \(60 ^ { \circ }\) anticlockwise about the origin followed by a one-way stretch in the \(x\)-direction, scale factor \(d ( d \neq 0 )\).

1. Find \(\mathbf { M }\) in terms of \(d\).
2. The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto a parallelogram of area \(\frac { 1 } { 2 } d ^ { 2 }\) units \({ } ^ { 2 }\). Show that \(d = 2\).
   The matrix \(\mathbf { N }\) is such that \(\mathbf { M N } = \left( \begin{array} { l l } 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } \end{array} \right)\).
3. Find \(\mathbf { N }\).
4. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M N }\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } 2 & k & k \\ 5 & - 1 & 3 \\ 1 & 0 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \quad \mathbf { C } = \left( \begin{array} { r c c } 0 & 1 & 1 \\ - 1 & 2 & 0 \end{array} \right)$$ where \(k\) is a real constant.

1. Find \(\mathbf { C A B }\).
2. Given that \(\mathbf { A }\) is singular, find the value of \(k\).
3. Using the value of \(k\) from part (b), find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).

1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
   It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
3. Find \(\mathbf { M }\).
4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
   The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
2. Find, in terms of \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
3. Show that the line through the origin with gradient \(\frac { 1 } { k - 1 }\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-08_2715_31_106_2016} Let \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
3. Give full details of the transformation represented by \(\mathbf { M }\).
4. Find the matrix \(\mathbf { N }\) such that \(\mathbf { N M } = \mathbf { C A B }\).

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 11 & 1 & 8 \\ 0 & - 2 & 0 \\ - 16 & 1 & 13 \end{array} \right)$$

1. Show that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue.
2. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\). \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-08_2717_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-09_2726_33_97_22}
3. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

1. Given that

$$\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 3 \\ - 2 & 3 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & k \\ 0 & - 3 \\ 2 k & 2 \end{array} \right)$$ where \(k\) is a non-zero constant,

1. determine the matrix \(\mathbf { A B }\)
2. determine the value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)

$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ The matrix \(\mathbf { P }\) represents a geometrical transformation \(U\)

1. Describe \(U\) fully as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(240 ^ { \circ }\) anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6
2. Determine the matrix \(\mathbf { Q }\), giving each entry in exact numerical form. Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\)
3. determine the matrix \(\mathbf { R }\) (ii) The transformation \(W\) is represented by the matrix $$\left( \begin{array} { c c } - 2 & 2 \sqrt { 3 } \\ 2 \sqrt { 3 } & 2 \end{array} \right)$$ Show that there is a real number \(\lambda\) for which \(W\) maps the point \(( \lambda , 1 )\) onto the point ( \(4 \lambda , 4\) ), giving the exact value of \(\lambda\) \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHI NI JLIYM ION OC
   VEYV SIHI NI ELIYM ION OC

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

$$\left( \begin{array} { r r } 5 & - 1 \\ 4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n \\ 4 n & 3 - 2 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$ is divisible by 7

3. $$\mathbf { A } = \left( \begin{array} { l l } 4 & - 2 \\ a & - 3 \end{array} \right)$$ where \(a\) is a real constant and \(a \neq 6\)

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { A } + 2 \mathbf { A } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

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

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 \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.

2. Given that $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 2 \\ - 1 & 0 & 5 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } 2 & 4 \\ - k & 2 k \\ 3 & 0 \end{array} \right) , \text { where } k \text { is a constant }$$

1. find the matrix \(\mathbf { A B }\),
2. find the exact value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)

8. (i) The transformation \(U\) is represented by the matrix \(\mathbf { P }\) where, $$P = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$

1. Describe fully the transformation \(U\). The transformation \(V\), represented by the matrix \(\mathbf { Q }\), is a stretch scale factor 3 parallel to the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Transformation \(U\) followed by transformation \(V\) is a transformation which is represented by matrix \(\mathbf { R }\).
3. Find the matrix \(\mathbf { R }\).
   (ii) $$S = \left( \begin{array} { r r } 1 & - 3 \\ 3 & 1 \end{array} \right)$$ Given that the matrix \(\mathbf { S }\) represents an enlargement, with a positive scale factor and centre \(( 0,0 )\), followed by a rotation with centre \(( 0,0 )\),
   1. find the scale factor of the enlargement,
   2. find the angle and direction of rotation, giving your answer in degrees to 1 decimal place.

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

1. Describe fully the transformations described by each of the matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\). It is given that the matrix \(\mathbf { D } = \mathbf { C A }\), and that the matrix \(\mathbf { E } = \mathbf { D B }\).
2. Find \(\mathbf { D }\).
3. Show that \(\mathbf { E } = \left( \begin{array} { c c } - 3 & 3 \\ 3 & 3 \end{array} \right)\). The triangle \(O R S\) has vertices at the points with coordinates \(( 0,0 ) , ( - 15,15 )\) and \(( 4,21 )\). This triangle is transformed onto the triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation described by \(\mathbf { E }\).
4. Find the coordinates of the vertices of triangle \(O R ^ { \prime } S ^ { \prime }\).
5. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\) and deduce the area of triangle \(O R S\).

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

1. Describe fully the geometrical transformation represented by the matrix \(\mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(A\) with coordinates \(( p , q )\) onto the point \(B\) with coordinates \(( 3 \sqrt { } 2,4 \sqrt { } 2 )\).
2. Find the value of \(p\) and the value of \(q\).
3. Find, in its simplest surd form, the length \(O A\), where \(O\) is the origin.
4. Find \(\mathbf { M } ^ { 2 }\). The point \(B\) is mapped onto the point \(C\) by the transformation represented by \(\mathbf { M } ^ { 2 }\).
5. Find the coordinates of \(C\).

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).

2.

1. $$\mathbf { A } = \left( \begin{array} { c c } - 4 & 10 \\ - 3 & k \end{array} \right) , \quad \text { where } k \text { is a constant. }$$ The triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { A }\). Given that the area of triangle \(T ^ { \prime }\) is twice the area of triangle \(T\), find the possible values of \(k\).
2. Given that $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 2 & 3 \\ - 2 & 5 & 1 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { r r } 2 & 8 \\ 0 & 2 \\ 1 & - 2 \end{array} \right)$$ find \(\mathbf { B C }\). \includegraphics[max width=\textwidth, alt={}, center]{9093bb1d-4f32-44e7-b0e7-b8c4f8a844e1-05_124_42_2608_1902}

5. \(\mathbf { R } = \left( \begin{array} { l l } a & 2 \\ a & b \end{array} \right)\), where \(a\) and \(b\) are constants and \(a > 0\).

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). Given that \(\mathbf { R } ^ { 2 }\) represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
2. find the value of \(a\) and the value of \(b\).

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

1. find \(\mathbf { A } ^ { 2 }\),
2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
   (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
   (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12 \\ k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.

2. (a) Given that $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 1 & 3 \\ 4 & 5 & 5 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & 1 \\ 1 & 2 \\ 0 & - 1 \end{array} \right)$$ find \(\mathbf { A B }\).
(b) Given that $$\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 8 & 6 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { r r } 5 & 2 k \\ 4 & k \end{array} \right) , \text { where } k \text { is a constant }$$ and $$\mathbf { E } = \mathbf { C } + \mathbf { D }$$ find the value of \(k\) for which \(\mathbf { E }\) has no inverse.

9. $$\mathbf { M } = \left( \begin{array} { r r } 3 & 4 \\ 2 & - 5 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(S ( 2 a - 7 , a - 1 )\), where \(a\) is a constant, onto the point \(S ^ { \prime } ( 25 , - 14 )\).
2. Find the value of \(a\). The point \(R\) has coordinates \(( 6,0 )\). Given that \(O\) is the origin,
3. find the area of triangle \(O R S\). Triangle \(O R S\) is mapped onto triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\). Given that $$\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$$
5. describe fully the single geometrical transformation represented by \(\mathbf { A }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by \(\mathbf { M }\).
6. Find B.

9. With reference to a fixed origin \(O\) and coordinate axes \(O x\) and \(O y\), a transformation from \(\mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(A\) where $$A = \left( \begin{array} { c c } 3 & 1 \\ 1 & - 2 \end{array} \right)$$

1. Find \(\mathrm { A } ^ { 2 }\).
2. Show that the matrix A is non-singular.
3. Find \(\mathrm { A } ^ { - 1 }\). The transformation represented by matrix A maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 1,2 - k )\), where \(k\) is a constant,
4. show that \(P\) lies on the line with equation \(y = 4 x - 1\)

8. $$\mathbf { A } = \left( \begin{array} { c c } 6 & - 2 \\ - 4 & 1 \end{array} \right)$$ and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Prove that $$\mathbf { A } ^ { 2 } = 7 \mathbf { A } + 2 \mathbf { I }$$
2. Hence show that $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 2 } ( \mathbf { A } - 7 \mathbf { I } )$$ The transformation represented by \(\mathbf { A }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( 2 k + 8 , - 2 k - 5 )\), where \(k\) is a constant,
3. find, in terms of \(k\), the coordinates of \(P\).

6. $$\mathbf { A } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { r r } - 1 & 1 \\ 0 & 1 \end{array} \right)$$ Given that \(\mathbf { M } = ( \mathbf { A } + \mathbf { B } ) ( 2 \mathbf { A } - \mathbf { B } )\),

1. calculate the matrix \(\mathbf { M }\),
2. find the matrix \(\mathbf { C }\) such that \(\mathbf { M C } = \mathbf { A }\).