Question #256463
Is matrix A consistent, motivate your answer in terms of eigenvalues. A = 2 1 0 0 2 0 2 3 1
1
Expert's answer
2021-10-26T08:55:22-0400

If AA is an n×nn × n matrix, then the sum of the nn eigenvalues of AA is the trace of AA and the product of the nn eigenvalues is the determinant of A.A.


A=(210020231)A=\begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0\\ 2 &3 & 1 \end{pmatrix}

AλI=(2λ1002λ0231λ)A-\lambda I=\begin{pmatrix} 2-\lambda & 1 & 0 \\ 0 & 2-\lambda & 0\\ 2 &3 & 1-\lambda \end{pmatrix}

det(AλI)=2λ1002λ0231λ\det (A-\lambda I)=\begin{vmatrix} 2-\lambda & 1 & 0 \\ 0 & 2-\lambda & 0\\ 2 &3 & 1-\lambda \end{vmatrix}

=(2λ)2λ031λ10021λ+002λ23=(2-\lambda)\begin{vmatrix} 2-\lambda & 0 \\ 3 & 1-\lambda \end{vmatrix}-1\begin{vmatrix} 0 & 0 \\ 2 & 1-\lambda \end{vmatrix}+0\begin{vmatrix} 0 & 2-\lambda \\ 2 & 3 \end{vmatrix}

=(2λ)(2λ)(1λ)=(2-\lambda)(2-\lambda)(1-\lambda)

Characteristic equation


det(AλI)=0\det (A-\lambda I)=0

(2λ)(2λ)(1λ)=0(2-\lambda)(2-\lambda)(1-\lambda)=0

λ1=1,λ2=2,λ3=2\lambda_1=1, \lambda_2=2, \lambda_3=2

These are the eigenvalues.

Hence


detA=λ1λ2λ3=1(2)(2)=40\det A=\lambda_1\lambda_2\lambda_3=1(2)(2)=4\not=0

=> matrix AA consistent.



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