# Find the characteristic polynomial of the matrices begin{bmatrix}1 & 2&1 0 & 1&2-1&3&2 end{bmatrix}

Find the characteristic polynomial of the matrices
$\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 2\\ -1& 3& 2\end{array}\right]$
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Asma Vang
Step 1
Given:
Let $A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 2\\ -1& 3& 2\end{array}\right]$
To Find: Characteristic polynomial of the given matrices
Step 2
Solution:
$A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 2\\ -1& 3& 2\end{array}\right]$
$A-\lambda I=\left[\begin{array}{ccc}1-\lambda & 2& 1\\ 0& 1-\lambda & 2\\ -1& 3& 2-\lambda \end{array}\right]$
Characteristic polynomial of matrix A is given by $|A-\lambda I|=0$
$\left[\begin{array}{ccc}1-\lambda & 2& 1\\ 0& 1-\lambda & 2\\ -1& 3& 2-\lambda \end{array}\right]=0$
$\left(1-\lambda \right)\left[\left(1-\lambda \right)\left(2-\lambda \right)-6\right]-2\left[0+2\right]+1\left[0+\left(1-\lambda \right)\right]=0$
$\left(1-\lambda \right)\left(2-\lambda -2\lambda +{\lambda }^{2}-6\right)-4+1-\lambda =0$
After simplifying we get
${\lambda }^{3}-4{\lambda }^{2}+7=0$
Hence this is the required characteristic polynomial of given matrix.
Jeffrey Jordon