Subject: Applied Mathematics 2

Topic: Matrices

Difficulty: High

Find the characteristic equation of the matrix A given below and hence ,find the matrix represented by $A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$ , where $A = \begin{bmatrix} \ 2 & 1 & 1 \\ \ 0 & 1 & 0 \\ \ 1 & 1 & 2 \\ \end{bmatrix}$

Characteristic eqn |A- λI| =0

i.e. $ \begin{vmatrix} 1-λ&2&-2 \\ -1&3-λ&0 \\ 0&-2&1-λ \end{vmatrix} = 0 $

Expanding we get a cubic equation in λ, as $ λ^3 - 5λ^2 + 7λ - 3 =0$

By C-H theorem we have $A^3 - 5A^2 + 7A – 3I =0$ ……….(1)

To …