Properties of matrix operations

6 The set \(S\) consists of all non-singular \(2 \times 2\) real matrices \(\mathbf { A }\) such that \(\mathbf { A Q } = \mathbf { Q A }\), where $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$$

1. Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\).
2. State clearly the restriction on the value of \(a\) such that \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\) is in \(S\).
3. Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)

9 Matrices \(\mathbf { M }\) and \(\mathbf { N }\) are given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { r r } 1 & - 3 \\ 1 & 4 \end{array} \right)\).

1. Find \(\mathbf { M } ^ { - 1 }\) and \(\mathbf { N } ^ { - 1 }\).
2. Find \(\mathbf { M N }\) and \(( \mathbf { M N } ) ^ { - \mathbf { 1 } }\). Verify that \(( \mathbf { M N } ) ^ { - 1 } = \mathbf { N } ^ { - 1 } \mathbf { M } ^ { - 1 }\).
3. The result \(( \mathbf { P Q } ) ^ { - 1 } = \mathbf { Q } ^ { - 1 } \mathbf { P } ^ { - 1 }\) is true for any two \(2 \times 2\), non-singular matrices \(\mathbf { P }\) and \(\mathbf { Q }\). The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof. $$\begin{aligned} & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q } = \mathbf { I } \\ \Rightarrow & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q Q } \mathbf { Q } ^ { - 1 } = \mathbf { I Q } ^ { - 1 } \end{aligned}$$

4 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are \(2 \times 2\) non-singular matrices and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix, simplify $$\mathbf { B } ( \mathbf { A B } ) ^ { - 1 } \mathbf { A } - \mathbf { I } .$$

5 Given that \(\mathbf { A }\) and \(\mathbf { B }\) are non-singular square matrices, simplify $$\mathbf { A B } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) ^ { - 1 } .$$

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)\) where \(t\) is a constant.

1. Show that \(| \mathbf { A } | = | \mathbf { B } |\).
2. Verify that \(| \mathbf { A B } | = | \mathbf { A } \| \mathbf { B } |\).
3. Given that \(| \mathbf { A B } | = - 1\) explain what this means about the constant \(t\).

3

1. You are given two matrices, A and B, where $$\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } - 1 & 2 \\ 2 & - 1 \end{array} \right)$$ Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
2. You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where $$\mathbf { C } = \left( \begin{array} { r r r } 2 & 1 & 5 \\ 1 & 1 & 3 \\ - 1 & 2 & 2 \end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r } - 4 & 8 & - 2 \\ - 5 & 9 & - 1 \\ 3 & - 5 & 1 \end{array} \right)$$ Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
3. The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } \\ 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant. Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 0 & 2 \\ 2 & 0 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right]$$

1. Calculate the matrix \(\mathbf { A B }\).
2. Show that \(\mathbf { A } ^ { 2 }\) is of the form \(k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
3. Show that \(( \mathbf { A B } ) ^ { 2 } \neq \mathbf { A } ^ { 2 } \mathbf { B } ^ { 2 }\).

2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & a \\ - 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 2 & 1 \end{array} \right)\), where \(a\) is
a constant. a constant.

1. By multiplying out the matrices on both sides of the equation, verify that \(\mathbf { A } ( \mathbf { B C } ) = ( \mathbf { A B } ) \mathbf { C }\).
2. State the property of matrix multiplication illustrated by this result.

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2 \\ 2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1 \\ 1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.

1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
   1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
   2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
   3. Hence verify your answer to part (i).

4 Anika thinks that, for two square matrices \(\mathbf { A }\) and \(\mathbf { B }\), the inverse of \(\mathbf { A B }\) is \(\mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }\). Her attempted proof of this is as follows. $$\begin{aligned} ( \mathbf { A B } ) \left( \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \right) & = \mathbf { A } \left( \mathbf { B A } ^ { - 1 } \right) \mathbf { B } ^ { - 1 } \\ & = \mathbf { A } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) \mathbf { B } ^ { - 1 } \\ & = \left( \mathbf { A } \mathbf { A } ^ { - 1 } \right) \left( \mathbf { B B } ^ { - 1 } \right) \\ & = \mathbf { I } \times \mathbf { I } \\ & = \mathbf { I } \\ \text { Hence } ( \mathbf { A B } ) ^ { - 1 } & = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \end{aligned}$$

1. Explain the error in Anika's working.
2. State the correct inverse of the matrix \(\mathbf { A B }\) and amend Anika's working to prove this.

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]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows. $$\mathbf{A} = \begin{bmatrix} p - 2 & p - 1 \\ 0 & 1 \end{bmatrix} \quad\quad \mathbf{B} = \begin{bmatrix} 1 & 2p - 1 \\ 0 & 4 - p \end{bmatrix}$$ Find the values of \(p\) such that \(\mathbf{A}\) and \(\mathbf{B}\) are commutative under matrix multiplication. Fully justify your answer. [4 marks]

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]

Matrices A and B are given by \(\mathbf{A} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} \frac{5}{13} & -\frac{12}{13} \\ \frac{12}{13} & \frac{5}{13} \end{pmatrix}\). Use A and B to disprove the proposition: "Matrix multiplication is commutative". [2]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find
   [3]

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]