Find the characteristic equation of the matrix A = $\begin{bmatrix} 1&3&7 \\ 4&2&3 \\ 1&2&1 \end{bmatrix} $ & hence find the matrix represented by $A^7 – 4A^6 – 20A^5 – 34A^4 - 4A^3 – 20A^2 – 33A + I $

Subject: Applied Mathematics 4

Topic: Matrix Theory

Difficulty: Medium

For characteristic equation |A - $\lambda$ I| = 0

$$ \begin{vmatrix} 1&3&7 \\ 4&2&3 \\ 1&2&1 \end{vmatrix} = 0 $$ $\lambda^3$ - (Sum of diagonal elements)$\lambda^2$ + (Sum of Principal minors)$\lambda$ - |A| = 0

$ \lambda^3 - (1+2+1)\lambda^2 + (-4-6-10)\lambda - 35 = 0 \\ \lambda^3 - 4 \lambda^2 + 20 \lambda - 35 = 0 $

By Cayley-Hamilton theorem

$ A^3 - 4 A^2 + 20 A - 35I = 0 $

Consider, $ A^7 -4A^6 -20A^5 - 34A^4 - 4A^3 - 20A^2 - 33A + I $

$ = A^4(A^3 - 4A^2 - 20A - 34I) - 4A^3 - 20A^2 - 33A + I \\ = A^4(A^3 - 4A^2 - 20A - 35I + I) + A^4 - 4A^3 - 20A^2 - 33A + I \\ = A^4(0) + A^4 - 4A^3 - 20A^2 - 33A + I \\ = A(A^3 - 4A^2 - 20A - 33I) + I \\ = A(A^3 - 4A^2 - 20A - 35I) + 2A + I \\ = A(0) + 2A + I \\ = 2A + I \\ = 2 \begin{bmatrix} 1&3&7 \\ 4&2&3 \\ 1&2&1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ = \begin{bmatrix} 3 & 6 & 14 \\ 8 & 5 & 6 \\ 2 & 4 & 3 \end{bmatrix} $