Given matrix is A = [ 2 1 1 0 1 0 1 1 2 ] A = \begin{bmatrix}
2 & 1 & 1 \\
0 & 1 & 0 \\
1 & 1 & 2
\end{bmatrix} A = ⎣ ⎡ 2 0 1 1 1 1 1 0 2 ⎦ ⎤
Then according to Cayley Hamilton theorem,
∣ A − λ I ∣ = 0 |A - \lambda I| = 0 ∣ A − λ I ∣ = 0
So we will have,
A − λ I = [ 2 1 1 0 1 0 1 1 2 ] − λ [ 1 0 0 0 1 0 0 0 1 ] A - \lambda I = \begin{bmatrix}
2 & 1 & 1 \\
0 & 1 & 0 \\
1 & 1 & 2
\end{bmatrix} - \lambda\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix} A − λ I = ⎣ ⎡ 2 0 1 1 1 1 1 0 2 ⎦ ⎤ − λ ⎣ ⎡ 1 0 0 0 1 0 0 0 1 ⎦ ⎤
∣ A − λ I ∣ = ∣ 2 − λ 1 1 0 1 − λ 0 1 1 2 − λ ∣ = 0 |A - \lambda I| = \begin{vmatrix}
2-\lambda & 1 & 1 \\
0 & 1-\lambda & 0 \\
1 & 1 & 2-\lambda
\end{vmatrix} = 0 ∣ A − λ I ∣ = ∣ ∣ 2 − λ 0 1 1 1 − λ 1 1 0 2 − λ ∣ ∣ = 0
Then equation will be
λ 3 − 5 λ 2 + 7 λ − 3 = 0 \lambda^3 - 5\lambda ^2+7\lambda -3=0 λ 3 − 5 λ 2 + 7 λ − 3 = 0
According to Cayley Hamilton theorem,
Every matrix is the root of it's eigen matrix.
then, A 3 − 5 A 2 + 7 A − 3 = 0 A^3 - 5A^2+7A -3=0 A 3 − 5 A 2 + 7 A − 3 = 0 (1)
Given equation is A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I A^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I
This equation can be written as,
A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I = ( A 3 − 5 A 2 + 7 A − 3 ) ( A 5 + A ) + ( A 2 + A + I ) 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) A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I = ( A 3 − 5 A 2 + 7 A − 3 ) ( A 5 + A ) + ( A 2 + A + I )
From equation (1), above equation will be modified as,
A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I = A 2 + A + I A^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I = A^2+A+I A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I = A 2 + A + I
So putting value of A 2 , A , I A^2,A,I A 2 , A , I
we will get,
A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I A^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I A 8 − 5 A 7 + 7 A 6 − 3 A 5 + A 4 − 5 A 3 + 8 A 2 − 2 A + I
= [ 5 4 4 0 1 0 4 4 5 ] + [ 2 1 1 0 1 0 1 1 2 ] + [ 1 0 0 0 1 0 0 0 1 ] = [ 8 5 5 0 3 0 5 5 8 ] = \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} = ⎣ ⎡ 5 0 4 4 1 4 4 0 5 ⎦ ⎤ + ⎣ ⎡ 2 0 1 1 1 1 1 0 2 ⎦ ⎤ + ⎣ ⎡ 1 0 0 0 1 0 0 0 1 ⎦ ⎤ = ⎣ ⎡ 8 0 5 5 3 5 5 0 8 ⎦ ⎤
Comments