Matrix inverse calculation

6. $$\mathbf { A } = \left( \begin{array} { r r r } x & 1 & 3 \\ 2 & 4 & x \\ - 4 & - 2 & - 1 \end{array} \right)$$

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

5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4 \\ 3 & 2 & 5 \\ 4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2 \\ 14 & - 14 & 7 \\ - 5 & 7 & - 4 \end{array} \right)\).

1. Calculate AB.
2. Write down \(\mathbf { A } ^ { - 1 }\).

\(\mathbf { 9 }\) The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 2 & 9 \\ 2 & a & 3 \\ 1 & 0 & - 1 \end{array} \right)\).

1. Find the determinant of \(\mathbf { X }\) in terms of \(a\).
2. Hence find the values of \(a\) for which \(\mathbf { X }\) is singular.
3. Given that \(\mathbf { X }\) is non-singular, find \(\mathbf { X } ^ { - 1 }\) in terms of \(a\).

6 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r r } a & 1 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)\), where \(a \neq 1\). Find \(\mathbf { C } ^ { - 1 }\).

4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)\).

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 \(a\).

5. $$\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 3 \\ 2 & 3 & 0 \\ 4 & a & 2 \end{array} \right) \quad \text { where } a \text { is a constant }$$

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

$$\mathbf { M } = \left( \begin{array} { c c c } 2 & a & 4 \\ 1 & - 1 & - 1 \\ - 1 & 2 & - 1 \end{array} \right)$$ 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 \(n\), $$\left( \begin{array} { l l } 3 & 0 \\ 6 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & 0 \\ 3 \left( 3 ^ { n } - 1 \right) & 1 \end{array} \right)$$

$$\mathbf{M} = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$ where \(p\) is a real constant

1. Find the exact values of \(p\) for which \(\mathbf{M}\) has no inverse. [4]

Given that \(\mathbf{M}\) does have an inverse,

1. find \(\mathbf{M}^{-1}\) in terms of \(p\). [5]

$$\mathbf{A} = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{pmatrix}, \quad u \neq 1.$$

1. Show that \(\det \mathbf{A} = 2(u - 1)\). [2]
2. Find the inverse of \(\mathbf{A}\). [6]

The image of the vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) when transformed by the matrix \(\begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & 6 \end{pmatrix}\) is \(\begin{pmatrix} 3 \\ 1 \\ 6 \end{pmatrix}\).

1. Find the values of \(a\), \(b\) and \(c\). [3]