4.03l Singular/non-singular matrices

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } k & 0 & 2 \\ 0 & - 1 & - 1 \\ 1 & 1 & - k \end{array} \right)$$ where \(k\) is a real constant.

1. Show that \(\mathbf { A }\) is non-singular.
   The matrices \(\mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 3 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 3 & - 1 & 1 \\ 1 & 1 & 2 \end{array} \right)$$ It is given that \(\mathbf { C A B } = \left( \begin{array} { l l } - 2 & - \frac { 3 } { 2 } \\ - 1 & - \frac { 3 } { 2 } \end{array} \right)\).
2. Find the value of \(k\).
3. 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 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 \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k \\ 0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.

1. Describe fully the other transformation and state the order in which the transformations are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).

1 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } k & 1 & 0 \\ 6 & 5 & 2 \\ - 1 & 3 & - k \end{array} \right)$$ where \(k\) is a real constant.

1. Show that \(\mathbf { A }\) is non-singular.
2. Given that \(\mathbf { A } ^ { - 1 } = \left( \begin{array} { c c c } 3 & 0 & - 1 \\ 1 & 0 & 0 \\ - \frac { 23 } { 2 } & \frac { 1 } { 2 } & 3 \end{array} \right)\), find the value of \(k\).

3 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0\), where \(c\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\).
2. Show that \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }\).
3. Find the real value of \(c\) for which the matrix \(\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)\) is singular.

5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.

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 \(\mathbf { C A B }\).
3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
   Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).

8

1. Find the value of \(a\) for which the system of equations $$\begin{gathered} 3 x + a y = 0 \\ 5 x - y = 0 \\ x + 3 y + 2 z = 0 \end{gathered}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 0 & 0 \\ 5 & - 1 & 0 \\ 1 & 3 & 2 \end{array} \right)$$
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
3. Use the characteristic equation of \(\mathbf { A }\) to show that $$( \mathbf { A } + 6 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

1

1. Show that the system of equations $$\begin{array} { r } x + 2 y + 3 z = 1 \\ 4 x + 5 y + 6 z = 1 \\ 7 x + 8 y + 9 z = 1 \end{array}$$ does not have a unique solution.
2. Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & - 6 a & 2 a + 2 \\ 0 & 1 - a & 0 \\ 0 & 2 - a & - 1 \end{array} \right)$$ where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).

1. Show that the equation \(\mathbf { A } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } 1 \\ 2 \\ 3 \end{array} \right)\) has a unique solution and interpret this situation geometrically.
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , 1 - a\) and - 1 .
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 4 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 4 }\) in terms of \(\mathbf { A }\) and \(a\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8

1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { c l } 6 x + a y & = 3 \\ 2 x - y & = 1 \\ x + 5 y + 4 z & = 2 \end{array}$$ has a unique solution.
2. Show that the system of equations in part (a) is consistent for all values of \(a\).
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0 \\ 2 & - 1 & 0 \\ 1 & 5 & 4 \end{array} \right)$$
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to show that $$( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

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 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13 \\ 0 & - 2 & 5 \\ 0 & 0 & 8 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

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

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 non-zero eigenvalues \(a , \frac { 1 } { 2 } , 2\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right) ,$$ respectively.
3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

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 \(k\).
   

1. $$\mathbf { M } = \left( \begin{array} { c c } 2 k + 1 & k \\ k + 7 & k + 4 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Show that \(\mathbf { M }\) is non-singular for all real values of \(k\).
2. Determine \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).

1. (i) \(\mathbf { A } = \left( \begin{array} { c c } - 3 & 8 \\ - 3 & k \end{array} \right) \quad\) where \(k\) is a constant The transformation represented by \(\mathbf { A }\) transforms triangle \(T\) to triangle \(T ^ { \prime }\) The area of triangle \(T ^ { \prime }\) is three times the area of triangle \(T\)

Determine the possible values of \(k\) (ii) \(\mathbf { B } = \left( \begin{array} { r r } a & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1 \\ 1 & 4 & 2 \end{array} \right)\) where \(a\) is a constant Determine, in terms of \(a\), the matrix \(\mathbf { C }\)

1. 1. The matrix \(\mathbf { A }\) is defined by

$$\mathbf { A } = \left( \begin{array} { c c } 3 k & 4 k - 1 \\ 2 & 6 \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
   (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0 \\ 0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
   State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
3. an enlargement about the origin with scale factor - 2
4. a reflection in the \(y\)-axis.

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

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

6. \(\mathbf { X } = \left( \begin{array} { l l } 1 & a \\ 3 & 2 \end{array} \right)\), where \(a\) is a constant.

1. Find the value of \(a\) for which the matrix \(\mathbf { X }\) is singular. $$\mathbf { Y } = \left( \begin{array} { r r } 1 & - 1 \\ 3 & 2 \end{array} \right)$$
2. Find \(\mathbf { Y } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) maps the point \(A\) onto the point \(B\).
   Given that \(B\) has coordinates ( \(1 - \lambda , 7 \lambda - 2\) ), where \(\lambda\) is a constant,
3. find, in terms of \(\lambda\), the coordinates of point \(A\).

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.

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.

1. $$\mathbf { M } = \left( \begin{array} { c c } x & x - 2 \\ 3 x - 6 & 4 x - 11 \end{array} \right)$$ Given that the matrix \(\mathbf { M }\) is singular, find the possible values of \(x\).