The eigen values of the matrix given below are \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 0&{ – 3}&{ – 4} \end{array}} \right]\] A. (0, -1, -3) B. (0, -2, -3) C. (0, 2, 3) D. (0, 1, 3)

The correct answer is $\boxed{\text{(0, -1, -3)}}$.

To find the eigenvalues of a matrix, we can use the following formula:

$$\lambda = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the characteristic polynomial of the matrix.

The characteristic polynomial of the matrix $A$ is given by:

$$p(x) = |A – xI| = x^3 – 4x^2 + 3x – 0$$

To find the eigenvalues of $A$, we need to solve the equation $p(x) = 0$.

Solving $p(x) = 0$, we get the following eigenvalues:

$$\lambda = 0, -1, -3$$

Therefore, the eigenvalues of the matrix $A$ are $\boxed{\text{(0, -1, -3)}}$.

Here is a brief explanation of each option:

• Option A: $(0, -1, -3)$. This is the correct answer.
• Option B: $(0, -2, -3)$. This is not the correct answer because the eigenvalue $-2$ is not an eigenvalue of the matrix $A$.
• Option C: $(0, 2, 3)$. This is not the correct answer because the eigenvalue $2$ is not an eigenvalue of the matrix $A$.
• Option D: $(0, 1, 3)$. This is not the correct answer because the eigenvalue $1$ is not an eigenvalue of the matrix $A$.