Answer:-
We use the fact that characteristics polynomial of matrix A given by
P(X)=det(A−λ1)=(λ1−X)(λ2−X).....(λn−X)
We know that if A is n×n matrix and λiiλiiλi is an Eigenvalue of A. It is a root of characteristics polynomial of A that is given by P(X)=det(A−Xl)=0
In =n×n identity matrix
P(λ1)=0
If λ1,λ2.......λn are Eigenvalue of matrix A then
P(λ1)=0 which is a monic polynomial
P(X)=(λ1−X)(λ2−X)........(λn−X)det(A−XIn)=(λ1−X)(λ2−X)........(λn−X)
Put X=0
We get
det(A)=(λ1−0)(λ2−0).......(λn−0)
=λ1,λ2......λn
det(A)=(λ1,λ2)......(λn)
Hence proved.
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