GO Classes DPP | GATE CS, DA | Linear Algebra | Matrix Operations and Eigen Values

Answer 1

Statement I is false, since row operations do not preserve eigenvalues in general.

For example,
$M=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
has eigenvalues $1,1$.

After interchanging the two rows, we get
$M'=\begin{pmatrix}0&1\\1&0\end{pmatrix}$,
whose eigenvalues are $1,-1$.

So Statement I is false.

Statement II is also false, since row operations on $(M-\lambda I)$ do not preserve eigenvalues in general.

Take
$M=\begin{pmatrix}1&0\\0&2\end{pmatrix}$ and $\lambda=1$.

Then
$M-\lambda I=\begin{pmatrix}0&0\\0&1\end{pmatrix}$,
whose eigenvalues are $0,1$.

After interchanging the two rows, we get
$(M-\lambda I)'=\begin{pmatrix}0&1\\0&0\end{pmatrix}$,
whose eigenvalues are $0,0$.

So Statement II is false.

Hence, the correct answer is $\boxed{\text{D. None}}$.

Answer 2

Therefore Option D is correct : None

Answer 3

Row operations do not preserve eigenvalues.

Answer: D

Answer 4

According to CH theorem, every matrix satisfies it's own characterstic equation. So, if the elements of the matrix chage after row operation, the characteristic equation also changes. And therefore, the eigen values will always be different.