Linear Algebra: Matrix operations

Question

Consider two statements:-

1) Eigenvalue of M = Eigenvalue of M' {M' is the matrix gets after row operation}

2) Eigenvalue of |M- $\lambda$I| = Eigenvalue of |M- $\lambda$I|' {|M- $\lambda$I|' is the matrix gets after row operation}

Answer 1

Both statements are false . If you are applying Row transformation on a matrix then you may get different eigen value.
Note : To answer such type of question take a 2*2 Matrix (Prefer element as 0 or 1) and apply row transformation and find its eigen value .. You will be get different eigen value in both cases

Comments

For second statement:  http://gateoverflow.in/2013 in this question for finding the Eigenvalues of the Matrix first we did $|A-\lambda I |$ and then apply the row operation. So, according to your 2nd statement both will not have same eigenvalues, which is contradicting the reference linked answer.

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There are tons of cases which is contradicting with this answer for example if i have scalar matrix then your first statement is TRUE no matter what row transformation you apply so there are various cases involved .

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@saxena0612 that is where I confused, but a true conclusion is always True under any set of values.

And to contradict any conclusion we can give the counter example.

https://gateoverflow.in/166952/maths-linear-algebra In this question also second statement of the question is holds true:-

2) Eigenvalue of |M- λλI| = Eigenvalue of |M- λλI|' {|M- λλI|' is the matrix gets after row operation}