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Let the matrix $A$ be given by: \begin{bmatrix} 2 & 0 \\ 2 & 3 \end{bmatrix} And let the matrices $B$, $C$, and $D$ be given by row operations: $R_2 \rightarrow R_2-R_1$, swap rows, and $R_2 \rightarrow -2R_2$, respectively. What are the eigenvalues of $A$,$B$, $C$, and $D$

I sorta guessed the rows and columns, for the transformation matrices. Is there a systematic way of doing this? Also can I compute these eigenvalues without evaluating $A$,$B$, $C$, and $D$?

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We have $\sigma(A) = \{2,3\}$ because the matrix is lower-triangular.

We have $B = \begin{bmatrix} 0 & 0 \\ 0 & 3\end{bmatrix}$ so $\sigma(B) = \{0,3\}$ because the matrix is diagonal.

We have $C = \begin{bmatrix} 2 & 3 \\ 2 & 0\end{bmatrix}$ so the characteristic polynomial is $x^2-2x-6$ so $\sigma(C) = \{1-\sqrt7, 1+\sqrt7\}$.

We have $D = \begin{bmatrix} 2 & 0 \\ -4 & -6\end{bmatrix}$ so $\sigma(D) = \{2,-6\}$ because the matrix is lower-triangular.

I don't know of an easy way to get the eigenvalues without calculating the matrices.

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