4.03j Determinant 3x3: calculation

4

1. A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 \\ 0 & 1 & - 2 \\ 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane. Find \(m\).
2. The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 \\ - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T . Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).

14 Three planes have equations $$\begin{aligned} k x - z & = 2 \\ - x + k y + 2 z & = 1 \\ 2 k x + 2 y + 3 z & = 0 \end{aligned}$$ where \(k\) is a constant.

1. By considering a suitable determinant, show that the three planes meet at a point for all values of \(k\).
2. Using a matrix method, find, in terms of \(k\), the coordinates of the point of intersection of the planes.

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8 \\ - 1 & 2 & 4 \\ 3 & 6 & 2 \end{array} \right)\).

1. In this question you must show detailed reasoning. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0\).
2. Use the Cayley-Hamilton theorem to determine \(\mathbf { A } ^ { - 1 }\). A matrix \(\mathbf { E }\) and a diagonal matrix \(\mathbf { D }\) are such that \(\mathbf { A } = \mathbf { E D E } ^ { - 1 }\). The elements in the diagonal of \(\mathbf { D }\) increase from top left to bottom right.
3. Determine the matrix \(\mathbf { D }\).

4 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8 \\ - 6 & 12 & 12 \\ - 2 & 4 & 8 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { P }\) is \(- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0\). You are given that the roots of this equation are 3,6 and 12 .
2. 1. Verify that \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) is an eigenvector of \(\mathbf { P }\), stating its associated eigenvalue.
   2. The vector \(\left( \begin{array} { l } x \\ y \\ z \end{array} \right)\) is an eigenvector of \(\mathbf { P }\) with eigenvalue 6. Given that \(z = 5\), find \(x\) and \(y\). You are given that \(\mathbf { P }\) can be expressed in the form \(\mathbf { E D E } ^ { - 1 }\), where \(\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1 \\ 1 & 2 & - 2 \\ 1 & 1 & 2 \end{array} \right)\) and \(\mathbf { D }\) is a diagonal matrix. The characteristic equation of \(\mathbf { E }\) is \(- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0\).
3. 1. Use the Cayley-Hamilton theorem to express \(\mathbf { E } ^ { - 1 }\) in terms of positive powers of \(\mathbf { E }\).
   2. Hence find \(\mathbf { E } ^ { - 1 }\).
   3. By identifying the matrix \(\mathbf { D }\) and using \(\mathbf { P } = \mathbf { E D E } ^ { - 1 }\), determine \(\mathbf { P } ^ { 4 }\).

1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { A } = \left( \begin{array} { r r } 3 & k \\ - 5 & 2 \end{array} \right)$$ where \(k\) is a constant.
Given that there exists a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix where $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { r r } 8 & 0 \\ 0 & - 3 \end{array} \right)$$

1. show that \(k = - 6\)
2. determine a suitable matrix \(\mathbf { P }\)

5. $$\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 3 \\ 2 & 3 & 0 \\ 4 & a & 2 \end{array} \right) \quad \text { where } a \text { is a constant }$$

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

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

1. Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that \(\operatorname { det } \mathbf { M } = 5\)
2. 1. find the value of \(k\)
   2. use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).

8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5 \\ 0 & 3 & p \\ - 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)

1. 1. determine the eigenvalue corresponding to this eigenvector
   2. hence show that \(p = 2\)
   3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
3. 1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u \\ v \\ w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x \\ y \\ z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u } \\ \dot { v } \\ \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right)\)
   2. determine a general solution for the displacement of the particle.

1. Matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 0 & a \\ - 3 & b & 1 \\ 0 & 1 & a \end{array} \right)$$ where \(a\) and \(b\) are integers, such that \(a < b\) Given that the characteristic equation for \(\mathbf { M }\) is $$\lambda ^ { 3 } - 7 \lambda ^ { 2 } + 13 \lambda + c = 0$$ where \(c\) is a constant,

1. determine the values of \(a , b\) and \(c\).
2. Hence, using the Cayley-Hamilton theorem, determine the matrix \(\mathbf { M } ^ { - 1 }\)

5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)\).

1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
2. Hence find the possible values of \(a\).
3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
5. Hence find the possible values of \(b\).

7 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .

1. In the case where \(x = 5\)

7 In this question you must show detailed reasoning. Matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } a & - 6 & a - 3 \\ a + 9 & a & 4 \\ 0 & - 13 & a - 1 \end{array} \right)\) where \(a\) is a constant.
Find all possible values of \(a\) for which \(\operatorname { det } \mathbf { A }\) has the same value as it has when \(a = 2\).

8 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1 \\ 1 - t & 6 & t \\ 2 - 2 t & 2 - 2 t & 1 \end{array} \right)\).

1. Find, in fully factorised form, an expression for \(\operatorname { det } \mathbf { A }\) in terms of \(t\).
2. State the values of \(t\) for which \(\mathbf { A }\) is singular. You are given the following system of equations in \(x , y\) and \(z\), where \(b\) is a real number. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5 \\ \left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10 \\ \left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
3. Determine which one of the following statements about the solution of the equations is true.

4 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .

1. In the case where \(x = 5\)

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

1. Show that \(\operatorname { det } \mathbf { A } = 6 - 3 a\).
2. State the value of \(a\) for which \(\mathbf { A }\) is singular.
3. Given that \(\mathbf { A }\) is non-singular find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

3

1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
2. State the value of \(k\) for which these three planes do not meet at a single point.
3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).

3

1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
2. State the value of \(k\) for which these three planes do not meet at a single point.
3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).

3

1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { r r r } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
2. State the value of \(k\) for which these three planes do not meet at a single point.
3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} 5 & -1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{pmatrix}.$$

1. Find the eigenvalues of \(\mathbf{A}\). [4]
2. Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\). [4]

The matrix A is given by $$\mathbf{A} = \begin{pmatrix} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{pmatrix}.$$

1. Find a matrix P and a diagonal matrix D such that \(\mathbf{A}^{-1} = \mathbf{PDP}^{-1}\). [7]
2. Use the characteristic equation of A to find \(\mathbf{A}^{-1}\). [4]

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{pmatrix}.$$

1. Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^3 - 12\lambda^2 + 124 + 80 = 0\) and find the eigenvalues of \(\mathbf{A}\). [4]
2. Use the characteristic equation of \(\mathbf{A}\) to show that $$\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A} + r\mathbf{I},$$ where \(p\), \(q\) and \(r\) are integers to be determined. [4]
3. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} - 3\mathbf{I})^4 = \mathbf{PDP}^{-1}\). [6]

$$\mathbf{M} = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$ where \(p\) is a real constant

1. Find the exact values of \(p\) for which \(\mathbf{M}\) has no inverse. [4]

Given that \(\mathbf{M}\) does have an inverse,

1. find \(\mathbf{M}^{-1}\) in terms of \(p\). [5]

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$

1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
3. Find a vector equation for the line \(l_1\). [5]

$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$

1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]

The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).

1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]