4.03o Inverse 3x3 matrix

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

$$\mathbf { M } = \left( \begin{array} { r r } 4 & 2 \\ 3 & - 1 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$$

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 \(\mathbf { P }\)

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

1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p \\ & 3 x - 6 y + 4 z = 1 \\ & 3 x + 2 y - z = 0 \end{aligned}$$
3. 1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1 \\ & 3 x - 5 y + 4 z = q \\ & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.

6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7 \\ 1 & 1 & 1 \\ 2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1 \\ x + y + z = p \\ 2 x + y - z = 2 \end{array}$$
3. 1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1 \\ x + y + z = q \\ 2 x + y - z = 2 \end{array}$$
   2. For this value of \(q\), determine a vector equation for the line of intersection.

8. $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & - 1 \\ 1 & 1 & 1 \\ k & 3 & 6 \end{array} \right) \quad k \neq 0$$

1. Find, in terms of \(k , \mathbf { A } ^ { - 1 }\)
2. Determine, in simplest form in terms of \(k\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 3 x + y - z = 3 \\ x + y + z = 1 \\ k x + 3 y + 6 z = 6 \end{array}$$

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

1. For which values of \(a\) does the matrix \(\mathbf { M }\) have an inverse? Given that \(\mathbf { M }\) is non-singular,
2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(a\) (ii) Prove by induction that for all positive integers \(n\), $$\left( \begin{array} { l l } 3 & 0 \\ 6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0 \\ 3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)$$

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

$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2 \\ - 2 & 3 & - 1 \\ 2 & - 1 & 3 \end{array} \right)$$

1. Show that 2 is a repeated eigenvalue of \(\mathbf { A }\) and find the other eigenvalue.
2. Hence find three non-parallel eigenvectors of \(\mathbf { A }\).
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix.

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 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1 \\ - 8 & 4 & 0 \\ 19 & - 8 & - 1 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence write down \(\mathbf { A } ^ { - 1 }\).
3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0 \\ 2 x + 5 y + 2 z = 1 \\ 3 x - 2 y - z = 4 \end{array}$$
   1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
   2. Find this unique solution.

5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$

1. Find the matrix:
   1. \(\mathbf { M } ^ { 2 }\);
   2. \(\mathbf { M } ^ { 4 }\).
2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
3. Find the matrix \(\mathbf { M } ^ { 2006 }\).

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 \(a\).

2 You are given the system of equations $$\begin{array} { r } a ^ { 2 } x - 2 y = 1 \\ x + b ^ { 2 } y = 3 \end{array}$$ where \(a\) and \(b\) are real numbers.

1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\).
2. Explain why the method used in part (a) works for all values of \(a\) and \(b\).

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

1. Prove by mathematical induction that, for all positive integers \(n\), $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l } 2 \times 3 ^ { n } & 0 \\ 3 ^ { n } - 1 & 2 \end{array} \right)$$
2. Find, in terms of \(n\), the inverse of \(\mathbf { A } ^ { n }\).

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

3 The points \(A ( 1,3 ) , B ( 4,36 )\) and \(C ( 9,151 )\) lie on the curve with equation \(y = p + q x + r x ^ { 2 }\).

1. Using this information, write down three simultaneous equations in \(p , q\) and \(r\).
2. Re-write this system of equations in the matrix form \(\mathbf { C x } = \mathbf { a }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, \(\mathbf { x }\) is an unknown vector, and \(\mathbf { a }\) is a fixed vector.
3. By finding \(\mathbf { C } ^ { - 1 }\), determine the values of \(p , q\) and \(r\).

2

1. Show that there is a value of \(t\) for which \(\mathbf { A B }\) is an integer multiple of the \(3 \times 3\) identity matrix \(\mathbf { I }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 2 & 1 \\ t & 1 & - t \\ 3 & 2 & 1 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { c r r } t - 2 & 0 & 5 \\ 12 & - 2 & - 6 \\ 3 t & 4 & 7 \end{array} \right) .$$
2. Express the system of equations $$\begin{aligned} - 5 x + 5 z & = 8 \\ 12 x - 2 y - 6 z & = 12 \\ - 9 x + 4 y + 7 z & = 22 \end{aligned}$$ in the form \(\mathbf { C x } = \mathbf { u }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, and \(\mathbf { x }\) and \(\mathbf { u }\) are suitable column vectors.
3. Use the result of part (i) to solve the system of equations given in part (ii).

9

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.

8

1. Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 5 a \\ 3 x + 8 y + a z & = b \end{aligned}$$ has a unique solution.
2. By use of the inverse matrix of a suitable \(3 \times 3\) matrix, find the unique solution in the case \(a = 1\) and \(b = 2\).
3. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.

The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).

1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]

The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).

1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]

Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8] Hence find a non-singular matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}\). [4]

It is given that the eigenvalues of the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$ are \(1, 3, 4\). Find a set of corresponding eigenvectors. [4] Write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$ where \(n\) is a positive integer. [2] Find \(\mathbf{P}^{-1}\) and deduce that $$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]

The matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 10 & -7 & 10 \\ 7 & -5 & 8 \end{pmatrix},$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. [3] It is given that \(\begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\). Find the corresponding eigenvalue. [2] Find a diagonal matrix \(\mathbf{D}\) and matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) such that \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D}\). [5]

The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).

1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
2. Find \(\mathbf{M}^T \mathbf{P}\). [3]