\text { We consider the matrix } A=\begin{array}{ccc} -1 & 0 & 0 \\ -1 & 2 & 0 \\ 0 & 4 & B \end{array} \text { Write the eigenvalues of } A \text { in ascending order (that is, } \lambda_{1} \leq \lambda_{2} \leq \lambda_{3} \text { ): } \text { (ii) Write the corresponding eigenvectors }\left(\vec{\nabla}_{1} \text { corresponds to } \lambda_{1}, \vec{V}_{2} \text { corresponds to } \lambda_{2}, \vec{\nabla}_{3} \text { corresponds to } \lambda_{3}\right. \text { ) } in their simplest form, such as the components indicated below are 1. Do not simplify any fractions that mightappear in your answers. (iii) Write the diagonalisation transformation X such that X^{-1} A X=\begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{array} and such that X has the following components equal to 1, x21 = X22=X33=1:

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