If A is an n×n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.
A=⎝⎛202123001⎠⎞
A−λI=⎝⎛2−λ0212−λ3001−λ⎠⎞
det(A−λI)=∣∣2−λ0212−λ3001−λ∣∣
=(2−λ)∣∣2−λ301−λ∣∣−1∣∣0201−λ∣∣+0∣∣022−λ3∣∣
=(2−λ)(2−λ)(1−λ) Characteristic equation
det(A−λI)=0
(2−λ)(2−λ)(1−λ)=0
λ1=1,λ2=2,λ3=2These are the eigenvalues.
Hence
detA=λ1λ2λ3=1(2)(2)=4=0 => matrix A consistent.
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