[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, \(\varphi(\lambda)=\lambda^{2}+1\)

Show that the following theorem is true: If two matrices are similar, then they have the same characteristic polynomial. Then show, by means of a counter-example, that the converse is false.

Let $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{4}+\mathrm{x}^{3}+4 \mathrm{x}^{2}+3 \mathrm{x}+1\( and \)\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+1$. Find the polynomials \(\mathrm{q}(\mathrm{x})\) and \(\mathrm{r}(\mathrm{x})\) such that $\mathrm{f}(\mathrm{x})=\mathrm{q}(\mathrm{x}) \mathrm{g}(\mathrm{x})+\mathrm{r}(\mathrm{x})$

Let \(\quad \varphi(\lambda)=-2-5 \lambda+3 \lambda^{2}\) $$ \begin{array}{rrrr}A= & \mid 1 & 2 \mid . & \text { Show that } & \varphi(A)=\mid 14 & 2 \mid \\ & \mid 3 & 1 \mid & & 13 & 14\end{array} $$

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) $\begin{array}{rrrr} & 2 & 0 & 0 & 0 \\ & \mid 0 & 2 & 0 & 0 \mid \\ & \mid 0 & 0 & 3 & 0 \\ & 0 & 0 & 0 & 3\end{array}$

Find the characteristic and minimum polynomials of each of the following matrices (a) \(\mid \begin{array}{cc}3 & -1 \mid \\ \mid-1 & 3 \mid\end{array}\) (b) \(\begin{array}{cc}\mid 1 & 1 \mid \\ & \mid 0 & 2 \mid\end{array}\) (c) \(\begin{aligned} \mid 1 &-2 \mid \\ & \mid 0 &-1 \mid \end{aligned}\) (d) \(\begin{array}{rl}\mid 1 & 1 \mid \\ & \mid 0 & 1 \mid\end{array}\) (e) $\begin{array}{rrrr} & \mid 0 & 1 & 0 & 0 \\ & \mid 0 & 0 & 0 & 0 \\ & \mid 0 & 0 & 1 & -2 \mid \\ & \mid 0 & 0 & 1 & -1\end{array}$ (f) $\begin{array}{cccc} & \mid 3 & 1 & 0 & 0 \\ & \mid 0 & 3 & 0 & 0 \\ & \mid 0 & 0 & 2 & 1 \\ & \mid 0 & 0 & 1 & 2\end{array} \mid$