Question

If A= [2 2 1] [1 3 1] [1 2 2] find the characteristic roots of A ( it is a 3×3 matrix)​

Answer 1

Answer

Characteristics polynomial

−λ³+7λ²−11λ+5

1 , 5

Find the detailed answer in picture

All the best

mark me brainliest and flw me

Answer 2

We need to recall the following definition of the characteristic roots.

The characteristic root of a matrix $A$  is $\lambda$ , where $|A-\lambda I|=0$ and $I$ is an identity matrix.

This problem is about the characteristic roots.

Given:

$A=\left[\begin{array}{ccc}2&2&1\\1&3&1\\1&2&2\end{array}\right]$

Let $\lambda$ be the characteristic root of a matrix $A$.

Then,

$|A-\lambda I|=0$

$|\left[\begin{array}{ccc}2&2&1\\1&3&1\\1&2&2\end{array}\right] -\lambda\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] |=0$

$|\left[\begin{array}{ccc}2&2&1\\1&3&1\\1&2&2\end{array}\right] -\left[\begin{array}{ccc}\lambda&0&0\\0&\lambda&0\\0&0&\lambda\end{array}\right] |=0$

$|\left[\begin{array}{ccc}2-\lambda&2&1\\1&3-\lambda&1\\1&2&2-\lambda\end{array}\right] |=0$

$(2-\lambda)[(3-\lambda)(2-\lambda)-2]-2[(2-\lambda)-1]+1[2-(3-\lambda)]=0$

$(2-\lambda)[6-5\lambda+\lambda^2-2]-2[1-\lambda]+1[-1+\lambda]=0$

$(2-\lambda)[\lambda^2-5\lambda+4]-2+2\lambda-1+\lambda=0$

$2\lambda^2-10\lambda+8-\lambda^3+5\lambda^2-4\lambda-2+2\lambda-1+\lambda=0$

$-\lambda^3+7\lambda^2-11\lambda+5=0$

$\lambda^3-7\lambda^2+11\lambda-5=0$

$(\lambda-1)^2(\lambda-5)=0$

$\lambda-1=0$  or  $\lambda-5=0$

$\lambda=1$  or  $\lambda=5$

Hence, the characteristic roots of a matrix $A$ are $1$, $1$, $5$.