(A) Define the characteristic polynomial of the matrix \(\mathrm{A}\), (B) Let $\mathrm{A}=\begin{array}{ccc}\mid 1 & 2 & -1 \mid \\ \mid 1 & 0 & 1 \mid \\ \mid 4 & -4 & 5\end{array}$ Find the characteristic polynomial of \(\mathrm{A}\).

Find the characteristic polynomials and the eigenvalues of the matrices. (i) \(\begin{array}{rl}\mathrm{A}= & 2 & 3 \\ & 1 & 4\end{array} \mid\) (ii) $\mathrm{B}=\mid \begin{array}{rr}\cos \alpha & \sin \alpha \\ \mid-\sin \alpha & \cos \alpha \mid\end{array}$ (iii) $\begin{array}{rlr}\mathrm{C}= & 1 & 2 & 3 \mid \\ \mid 2 & 1 & 3 \mid \\\ \mid 3 & 3 & 6\end{array} \mid$

Let \(\mathrm{V}\) be the vector space \(\mathrm{R}^{2}\) and let \(\mathrm{T}\) be the operator defined by $\quad \mathrm{T}(\mathrm{xy})=(2 \mathrm{x}-\mathrm{y}, \mathrm{x}+\mathrm{y})$ Let \(\mathrm{f}(\mathrm{x})=2+3 \mathrm{x}\) and \(\mathrm{g}(\mathrm{x})=\mathrm{x}+\mathrm{x}^{2} .\) Find \(\mathrm{f}(\mathrm{T})\) and \(\mathrm{g}(\mathrm{T})\)

[A] Find the minimum polynomial \(\mathrm{m}(\lambda)\) of the matrix $\mathrm{A}=\begin{array}{cccc}2 & 1 & 0 & 0 \\ & 0 & 2 & 0 & 0 \mid \\ & 0 & 0 & 2 & 0 \\ & 0 & 0 & 0 & 5\end{array}$ [B] Let \(\mathrm{A}\) be a 3 by 3 matrix over the real field cannot be a zero of the polynomial, \(\varphi(\lambda)=\lambda^{2}+1\)

Find the minimal polynomials of the following matrices: (i) \(\begin{array}{cc}\mid 3 & 1 \mid \\ \mid 0 & 3 \mid\end{array}\) (ii) $\begin{array}{ccc}\mid 3 & 1 & 0 \\ & \mid 0 & 3 & 0 \mid \\ & \mid 0 & 0 & 3 \mid\end{array}$ (iii) $\begin{array}{ccc}\mid 2 & 0 & 0 \\ & \mid 0 & 3 & 1 \mid \\ \mid 0 & 0 & 3 \mid\end{array}$ (iv) $\begin{array}{rrrr} & 2 & 0 & 0 & 0 \\ & \mid 0 & 2 & 0 & 0 \mid \\ & \mid 0 & 0 & 3 & 0 \\ & 0 & 0 & 0 & 3\end{array}$

If $\mathrm{F}(\mathrm{X})=\begin{array}{ccccc}1 & 0 & 1 \mid & \mid 2 & 1 & 0 \mid \\ & \mid 2 & 1 & 1 \mid & -\mid-1 & 1 & 1 \\ & 1 & 1 & 1 \mid & \mid 0 & 1 & 0\end{array} \mid$ $\begin{array}{rllll} & \mid 1 & 1 & 1 \mid & & \\ \mathrm{X}+ & \mid 1 & 0 & 1 \mid \mathrm{X}^{2}, & \text { and } \mathrm{B}= & 1 & 1 \\ & \mid 0 & 1 & 0 \mid & 1 & 0 \\ & \text { find } \mathrm{F}_{\mathrm{L}}(\mathrm{B}) & \text { and } \mathrm{F}_{\mathrm{R}}(\mathrm{B}) .\end{array}$