Matrix properties verification

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 \(k\).

2. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 3 & a \\ 2 & 0 & 1 \\ 1 & - 2 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 4 \\ 3 & - 2 & 3 \\ 1 & 2 & b \end{array} \right)$$ where \(a\) and \(b\) are constants.

1. Write down \(\mathbf { A } ^ { \mathrm { T } }\) in terms of \(a\).
2. Calculate \(\mathbf { A B }\), giving your answer in terms of \(a\) and \(b\).
3. Hence show that $$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$

3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1 \\ 2 & 0 \end{array} \right)\), where \(k\) is a constant.

1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.

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 \(b\).

3 The matrix \(\mathbf { A }\) is such that \(\operatorname { det } ( \mathbf { A } ) = 2\) Determine the value of \(\operatorname { det } \left( \mathbf { A } ^ { - 1 } \right)\) Circle your answer.
-2 \(- \frac { 1 } { 2 }\) \(\frac { 1 } { 2 }\) 2

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 0 & p \\ a & b & c \end{pmatrix},$$ where \(p\), \(a\), \(b\) and \(c\) are constants and \(a > 0\). Given that \(\mathbf{M}\mathbf{M}^T = k\mathbf{I}\) for some constant \(k\), find

1. the value of \(p\), [2]
2. the value of \(k\), [2]
3. the values of \(a\), \(b\) and \(c\), [6]
4. \(|\det \mathbf{M}|\). [2]