4.03n Inverse 2x2 matrix

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

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).

1

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4 \\ 2 & 2 \end{array} \right)\).
2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\).
4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).

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 \(\mathbf { M }\).

8

1. Find the value of \(a\) for which the system of equations $$\begin{array} { r } 13 x + 18 y - 28 z = 0 \\ - 4 x - a y + 8 z = 0 \\ 2 x + 6 y - 5 z = 0 \end{array}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 13 & 18 & - 28 \\ - 4 & - 1 & 8 \\ 2 & 6 & - 5 \end{array} \right)$$
2. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\) in terms of \(\mathbf { A }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 4 & 2 \\ 0 & - 1 & 1 \\ 0 & 0 & 2 \end{array} \right) .$$

1. State the eigenvalues of \(\mathbf { P }\).
2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
   The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues \(b , - 1,1\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right)$$ respectively.
3. Find \(\mathbf { A }\) in terms of b.

9 It is given that \(a\) is a positive constant.

1. Show that the system of equations $$\begin{aligned} a x + ( 2 a + 5 ) y + ( a + 1 ) z & = 1 \\ - 4 y & = 2 \\ 3 y - z & = 3 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & 2 a + 5 & a + 1 \\ 0 & - 4 & 0 \\ 0 & 3 & - 1 \end{array} \right)$$
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
3. Find a matrix \(\mathbf { P }\) such that $$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r } a & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 4 \end{array} \right) \mathbf { P } ^ { - 1 } .$$
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7 \\ 0 & 5 & 7 \\ 0 & 0 & - 2 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.

7

1. It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector \(\mathbf { e }\). Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 15 & - 4 & 3 \\ 3 & 0 & 2 \end{array} \right)$$
2. Given that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
3. It is also given that \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) are eigenvectors of \(\mathbf { A }\). Find the corresponding eigenvalues.
4. Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
5. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q l\), where \(p\) and \(q\) are rational numbers to be determined.

7. (i) $$\mathbf { A } = \left( \begin{array} { r r } 6 & k \\ - 3 & - 4 \end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$ Find, in terms of \(k\),

1. \(\mathbf { A } ^ { - 1 }\)
2. \(\mathbf { A } ^ { 2 }\) Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9 \\ - 3 & - 5 \end{array} \right)\)
3. find the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { c c } - \frac { 1 } { 2 } & - \sqrt { 3 } \\ \frac { \sqrt { 3 } } { 2 } & - 1 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\).
   1. Find the value of \(p\).
   2. Find the value of \(\theta\).

3. The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c } k + 5 & - 2 \\ - 3 & k \end{array} \right)$$

1. Determine the values of \(k\) for which \(\mathbf { M }\) is singular. Given that \(\mathbf { M }\) is non-singular,
2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).
   

$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ 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

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a \\ 4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

1. Find the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(a\).
   (3) The matrix \(\mathbf { P }\) represents the transformation \(U\) which transforms a triangle \(T _ { 1 }\) onto the triangle \(T _ { 2 }\).
   The triangle \(T _ { 2 }\) has vertices at the points ( \(- 3 a , - 4 a\) ), ( \(6 a , 8 a\) ), and ( \(- 20 a , 15 a\) ).
2. Find the coordinates of the vertices of \(T _ { 1 }\)
3. Hence, or otherwise, find the area of triangle \(T _ { 2 }\) in terms of \(a\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through an angle \(\alpha\) clockwise about the origin, where \(\tan \alpha = \frac { 4 } { 3 }\) and \(0 < \alpha < \frac { \pi } { 2 }\)
4. Write down the matrix \(\mathbf { Q }\), giving each element as an exact value. The transformation \(U\) followed by the transformation \(V\) is the transformation \(W\). The matrix \(\mathbf { R }\) represents the transformation \(W\).
5. Find the matrix \(\mathbf { R }\).

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

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).

4. $$\mathbf { A } = \left( \begin{array} { c c } 2 p & 3 q \\ 3 p & 5 q \end{array} \right)$$ where \(p\) and \(q\) are non-zero real constants.

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\) and \(q\). Given \(\mathbf { X A } = \mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { c c } p & q \\ 6 p & 11 q \\ 5 p & 8 q \end{array} \right)$$
2. find the matrix \(\mathbf { X }\), giving your answer in its simplest form.

1. $$\mathbf { A } = \left( \begin{array} { r r } 3 & a \\ - 2 & - 2 \end{array} \right)$$ where \(a\) is a non-zero constant and \(a \neq 3\)

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

1. Given that

$$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
   

5. \(\mathbf { A } = \left( \begin{array} { c c } a & - 5 \\ 2 & a + 4 \end{array} \right)\), where \(a\) is real.

1. Find \(\operatorname { det } \mathbf { A }\) in terms of \(a\).
2. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(a\). Given that \(a = 0\),
3. find \(\mathbf { A } ^ { - 1 }\).

8. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 2 \\ - 1 & 3 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { A }\).
2. Find \(\mathbf { A } ^ { - 1 }\). The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\). Given that the area of triangle \(S\) is 72 square units,
3. find the area of triangle \(R\). The triangle \(S\) has vertices at the points \(( 0,4 ) , ( 8,16 )\) and \(( 12,4 )\).
4. Find the coordinates of the vertices of \(R\).

8. $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 2 & 3 \end{array} \right)$$

1. Show that \(\mathbf { A }\) is non-singular.
2. Find \(\mathbf { B }\) such that \(\mathbf { B A } ^ { 2 } = \mathbf { A }\).

7. \(\mathbf { A } = \left( \begin{array} { r r } a & - 2 \\ - 1 & 4 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { A }\) is singular. $$\mathbf { B } = \left( \begin{array} { r r } 3 & - 2 \\ - 1 & 4 \end{array} \right)$$
2. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { B }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 6,3 k + 12 )\), where \(k\) is a constant,
3. show that \(P\) lies on the line with equation \(y = x + 3\).

2. \(\mathbf { M } = \left( \begin{array} { c c } 2 a & 3 \\ 6 & a \end{array} \right)\), where \(a\) is a real constant.

1. Given that \(a = 2\), find \(\mathbf { M } ^ { - 1 }\).
2. Find the values of \(a\) for which \(\mathbf { M }\) is singular.

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 \(\mathbf { M C } = \mathbf { A }\).