Let $A= \begin{pmatrix} 1 & 0 & 2 \\ 1 & -2 & 0 \\ 0 & 0 & - 3 \\ \end{pmatrix} $ and $I$ be the identity matrix. If $6A^-=aA^2+bA+cI$ where $a,b,c$ are real numbers, then obtain the value of $(a,b,c).$
Options are: a) (1,2,1), b) (1,-1,2), c) (4,1,1), d) (1,4,1)
Calculate this with proper explanation and minimal working please. I am new to this topic. Thanks in advance.
If $p(t)=a_3t^3+a_2t^2+a_1t+a_0$ is the char. polynomial of $A$, then , by Cayley -Hamilton:
$a_3A^3+a_2A^2+a_1A+a_0I=0$.
Compute $a_3,a_2,a_1$ and $a_0$. You will see thatb $a_0 \ne 0$.
This gives:
$a_0A^{-1}=a_3A^2+a_2A+a_1I$
Your turn !