Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 06

Year : MAY 2014

Since A is a upper triangular matrix, Eigen value (x)= diagonal element= $2,2,2$

Case 1: Let $f(x)=(x-2)\\ \therefore f(A)=A-2I$

$$A= \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{bmatrix}-2\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$

$$=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix}\neq 0$$

$\therefore f(x)=x-2$ does not annihilates A

So, f(x) in not a minimal polynomial

Case 2: Let $g(x)=(x-2)(x-2)=x^2-4x+4$

Now $A^2=A\times A$

$$=\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{bmatrix}\times \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{bmatrix}$$

$$=\begin{bmatrix} 4 & 4 & 0 \\ 0 & 4 & 4 \\ 0 & 0 & 4 \\ \end{bmatrix}$$

$$\therefore A^2-4A+4I$$

$$\begin{bmatrix} 4 & 4 & 0 \\ 0 & 4 & 4 \\ 0 & 0 & 4 \\ \end{bmatrix}-4\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \\ \end{bmatrix}+4\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$

$$\begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\neq 0$$

$\therefore g(x)=x^2-4x+4$ does not annihilates A

So, g(x) is not a minimal polynomial

Here, matrix A is not derogatory