- The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & 1 & a
2 & b & c
- 1 & 0 & 1
\end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$
- Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
- the values of \(a , b\) and \(c\),
- the eigenvalues which correspond to the two given eigenvectors.
- The matrix \(\mathbf { P }\) is given by
$$\mathbf { P } = \left( \begin{array} { r r r }
1 & 1 & 0
2 & 1 & d
- 1 & 0 & 1
\end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$
Find
- the determinant of \(\mathbf { P }\) in terms of \(d\),
- the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).