4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1)

5 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l } a & 0 \\ 2 & 3 \end{array} \right)\) where \(a\) is a constant and \(a \neq 3\).

1. Given that the acute angle between the directions of the eigenvectors of \(\mathbf { P }\) is \(\frac { 1 } { 4 } \pi\) radians, determine the possible values of \(a\).
2. You are given instead that \(\mathbf { P }\) satisfies the matrix equation \(\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }\) for some rational number \(r\).
   1. Use the Cayley-Hamilton theorem to determine the value of \(a\) and the corresponding value of \(r\).
   2. Hence show that \(\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }\) where \(s\) and \(t\) are rational numbers to be determined. You should not calculate \(\mathbf { P } ^ { 4 }\).

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 \(\mathbf { P } ^ { 4 }\).

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

1. determine the eigenvalue corresponding to this eigenvector.
2. Hence show that \(p = 3\)
3. Determine
   1. the remaining eigenvalues of \(\mathbf { A }\),
   2. corresponding eigenvectors for these eigenvalues.
4. Hence determine a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { \mathrm { T } }\)

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

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For each of the eigenvalues find a corresponding eigenvector.
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P }\) is a diagonal matrix.

4

1. Find the value of \(k\) for which matrix \(\mathbf { A }\) is singular. 4
2. Describe the transformation represented by matrix \(\mathbf { B }\). 4
3. (i) Given that \(\mathbf { A }\) and \(\mathbf { B }\) are both non-singular, verify that \(\mathbf { A } ^ { \mathbf { - 1 } } \mathbf { B } ^ { \mathbf { - 1 } } = ( \mathbf { B A } ) ^ { \mathbf { - 1 } }\).
   [0pt] [4 marks]
   4 (c) (ii) Prove the result \(\mathbf { M } ^ { - \mathbf { 1 } } \mathbf { N } ^ { - \mathbf { 1 } } = ( \mathbf { N M } ) ^ { - \mathbf { 1 } }\) for all non-singular square matrices \(\mathbf { M }\) and \(\mathbf { N }\) of the same size.
   [0pt] [4 marks]

\(\mathbf{A}\) and \(\mathbf{B}\) are non-singular square matrices.

1. Write down the product \(\mathbf{AA}^{-1}\) as a single matrix. [1 mark]
2. \(\mathbf{M}\) is a matrix such that \(\mathbf{M} = \mathbf{AB}\). Prove that \(\mathbf{M}^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) [3 marks]

\(\mathbf{A}\) is a non-singular \(2 \times 2\) matrix and \(\mathbf{A}^T\) is the transpose of \(\mathbf{A}\)

1. Using the result $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$ show that $$(\mathbf{A}^{-1})^T = (\mathbf{A}^T)^{-1}$$ [3 marks]
2. It is given that \(\mathbf{A} = \begin{pmatrix} 4 & 5 \\ -1 & k \end{pmatrix}\), where \(k\) is a real constant.
   1. Find \((\mathbf{A}^{-1})^T\), giving your answer in terms of \(k\) [2 marks]
   2. State the restriction on the possible values of \(k\) [1 mark]

A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"

1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
2. Refine the student's claim to make it fully correct. [1 mark]
3. Prove that your answer to part (b) is correct. [3 marks]

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.

1. Write down A. [1]
2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]

Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.

1. Describe T. [1]
2. Determine the matrix which represents the transformation S followed by T. [2]
3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]

In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the \(x\)-axis. A is the matrix which represents S.

1. Write down A. [1]
2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]

Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.

1. Describe T. [1]
2. Determine the matrix which represents the transformation S followed by T. [2]
3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]

For real values of \(t\), the non-singular matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$

1. Determine the values which \(t\) cannot take. [2]
2. Without finding either \(\mathbf{A}\) or \(\mathbf{B}\), determine \((\mathbf{AB})^{-1}\) in terms of \(t\). [2]