Show that A and B are not similar matrices A=begin{bmatrix}1 & 0 &1 2 & 0 &2 3&0&3 end{bmatrix} , B=begin{bmatrix}1 & 1 &0 2 & 2 &0 0&1&1 end{bmatrix}

Yulia 2020-12-06 Answered
Show that A and B are not similar matrices
A=[101202303],B=[110220011]
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Expert Answer

d2saint0
Answered 2020-12-07 Author has 89 answers
Step 1
If the matrices A and B are not similar, then their eigenvalues are not the same.
Step 2
Consider the matrix A and find its eigenvalues as follows.
A=[101202303]
|AλI|=0
|[101202303]λ[100010001]|=0
|1λ012λ2303λ|=0
(1λ)(λ)(3λ)0+1(0+3λ)=0
λ3+4λ2=0
λ(λ2+4λ)=0
λ(λ24λ)=0
λ(λ)(λ4)=0
λ1=0,λ2=0,λ3=4
Consider the matrix B and find its eigenvalues as follows.
B=[110220011]
|BλI|=0
|[110220011]λ[100010001]|=0
|1λ1022λ0011λ|=0
(1λ)(2λ)(1λ)1(2(1λ))+0=0
λ3+4λ23λ=0
λ(λ2+4λ3)=0
λ(λ24λ+3)=0
λ(λ1)(λ3)=0
λ1=0,λ2=1,λ3=3
Observe that the eigenvalues of the matrices A and B are not the same and thus these matrices are not similar.
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Jeffrey Jordon
Answered 2022-01-23 Author has 2087 answers

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