Non-singular matrix proof

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

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

4. Given that $$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).

1. Given that

$$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
   

5. \(\mathbf { A } = \left( \begin{array} { c c } a & - 5 \\ 2 & a + 4 \end{array} \right)\), where \(a\) is real.

1. Find \(\operatorname { det } \mathbf { A }\) in terms of \(a\).
2. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(a\). Given that \(a = 0\),
3. find \(\mathbf { A } ^ { - 1 }\).

9. With reference to a fixed origin \(O\) and coordinate axes \(O x\) and \(O y\), a transformation from \(\mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(A\) where $$A = \left( \begin{array} { c c } 3 & 1 \\ 1 & - 2 \end{array} \right)$$

1. Find \(\mathrm { A } ^ { 2 }\).
2. Show that the matrix A is non-singular.
3. Find \(\mathrm { A } ^ { - 1 }\). The transformation represented by matrix A maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( k - 1,2 - k )\), where \(k\) is a constant,
4. show that \(P\) lies on the line with equation \(y = 4 x - 1\)