Question #285045

(i) Using Cayley Hamilton theorem, find 𝐴




8 − 𝐴




7 + 5𝐴




6 − 𝐴




5 + 𝐴




4 − 𝐴




3 + 6𝐴




2 +




𝐴 − 2𝐼 𝑖𝑓 [




1 2 −2




2 5 −4




3 7 −5




]


1
Expert's answer
2022-01-10T13:00:37-0500

Given matrix is A=[211010112]A = \begin{bmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{bmatrix}


Then according to Cayley Hamilton theorem,

AλI=0|A - \lambda I| = 0


So we will have,


AλI=[211010112]λ[100010001]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}



AλI=2λ1101λ0112λ=0|A - \lambda I| = \begin{vmatrix} 2-\lambda & 1 & 1 \\ 0 & 1-\lambda & 0 \\ 1 & 1 & 2-\lambda \end{vmatrix} = 0




Then equation will be

λ35λ2+7λ3=0\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, A35A2+7A3=0A^3 - 5A^2+7A -3=0 (1)


Given equation is A85A7+7A63A5+A45A3+8A22A+IA^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I


This equation can be written as,

A85A7+7A63A5+A45A3+8A22A+I=(A35A2+7A3)(A5+A)+(A2+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)


From equation (1), above equation will be modified as,

A85A7+7A63A5+A45A3+8A22A+I=A2+A+IA^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I = A^2+A+I


So putting value of A2,A,IA^2,A,I

we will get,

A85A7+7A63A5+A45A3+8A22A+IA^8 − 5A^7 + 7A^6 − 3A^5 + A^4 − 5A^3 + 8A^2 − 2A + I

=[544010445]+[211010112]+[100010001]=[855030558]= \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}




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