Question

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

Matrices
ANSWERED
asked 2021-02-27
Find the characteristic polynomial of the matrices
\(\begin{bmatrix}1 & 2&1 \\0 & 1&2\\-1&3&2 \end{bmatrix}\)

Answers (1)

2021-02-28
Step 1
Given:
Let \(A=\begin{bmatrix}1 & 2&1 \\0 & 1&2\\-1&3&2 \end{bmatrix}\)
To Find: Characteristic polynomial of the given matrices
Step 2
Solution:
\(A=\begin{bmatrix}1 & 2&1 \\0 & 1&2\\-1&3&2 \end{bmatrix}\)
\(A-\lambda I=\begin{bmatrix}1-\lambda & 2&1 \\0 & 1-\lambda&2\\-1&3&2-\lambda \end{bmatrix}\)
Characteristic polynomial of matrix A is given by \(|A-\lambda I|=0\)
\(\begin{bmatrix}1-\lambda & 2&1 \\0 & 1-\lambda&2\\-1&3&2-\lambda \end{bmatrix}=0\)
\((1-\lambda)\left[(1-\lambda)(2-\lambda)-6\right]-2\left[0+2\right]+1\left[0+(1-\lambda)\right]=0\)
\((1-\lambda)(2-\lambda-2\lambda+\lambda^2-6)-4+1-\lambda=0\)
After simplifying we get
\(\lambda^3-4\lambda^2+7=0\)
Hence this is the required characteristic polynomial of given matrix.
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