Question

Find the characteristic equation of the matrix A =





2 1 1

0 1 0

1 1 2



 and hence find the matrix

represented by A8 − 5A7 + 7A6 − 3A5 + A4 − 5A3 + 8A2 − 2A + I.

Answer 1

Characteristic equation $\det(A-\lambda I)=0$

$$\begin{vmatrix} 2-\lambda & 1 & 1 \\ 0 & 1-\lambda & 0 \\ 1 & 1 & 2-\lambda \\ \end{vmatrix}=0$$

$$(2-\lambda)\begin{vmatrix} 1-\lambda & 0 \\ 1 & 2-\lambda \end{vmatrix}-\begin{vmatrix} 0 & 0 \\ 1 & 2-\lambda \end{vmatrix}+\begin{vmatrix} 0 & 1-\lambda \\ 1 & 1 \end{vmatrix}=0$$

$$(2-\lambda)^2(1-\lambda)-(1-\lambda)=0$$

$$(1-\lambda)(4-4\lambda+\lambda^2-1)=0$$

$$(1-\lambda)^2(3-\lambda)=0$$

$$\lambda_1=1, \lambda_2=1, \lambda_3=3$$

The equation can be written as

$$\lambda^3-5\lambda^2+7\lambda-3=0$$

According to Cayley Hamilton theorem, every matrix is the root of it's eigen matrix. Then

$$A^3-5A^2+7A-3=0$$

Given sum

This sum can be written as,

$$A^8 − 5A^7 + 7A^6 − 3A^5$$

$$+ A^4 − 5A^3 + 8A^2 − 2A + I$$

$$= (A^3 - 5A^2+7A -3)(A^5+A) + (A^2+A+I)$$

Since $A^3-5A^2+7A-3=0,$ then

$$A^8 − 5A^7 + 7A^6 − 3A^5$$

$$+ A^4 − 5A^3 + 8A^2 − 2A + I$$

$$=A^2+A+I$$

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

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

$$=\begin{bmatrix} 5 & 4 & 4 \\ 0 & 1 & 0 \\ 4 & 4 & 5 \\ \end{bmatrix}$$

$$A^8 − 5A^7 + 7A^6 − 3A^5$$

$$+ A^4 − 5A^3 + 8A^2 − 2A + I$$

$$=A^2+A+I$$

$$=\begin{bmatrix} 5 & 4 & 4 \\ 0 & 1 & 0 \\ 4 & 4 & 5 \\ \end{bmatrix}+\begin{bmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \\ \end{bmatrix}+\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$

$$=\begin{bmatrix} 8 & 5 & 5 \\ 0 & 3 & 0 \\ 5 & 5 & 8 \\ \end{bmatrix}$$