What are the eigen values of the following 2 × 2 matrix? \[\left[ {\begin{array}{*{20}{c}} 2&{ – 1} \\ { – 4}&5 \end{array}} \right]\] A. -1 and 1 B. 1 and 6 C. 2 and 5 D. 4 and -1

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a 2×2 matrix can be computed as follows:

$$|A – \lambda I| = \det \left[ {\begin{array}{{20}{c}} a&b \ c&d \end{array}} – \lambda \begin{array}{{20}{c}} 1&0 \ 0&1 \end{array}} \right] = \lambda ^2 – (a+d) \lambda + ad – bc$$

In this case, $a=2$, $b=-1$, $c=-4$, and $d=5$. Substituting these values into the characteristic polynomial formula, we get:

$$\lambda ^2 – (2+5) \lambda + 2 \cdot 5 – (-1) \cdot (-4) = \lambda ^2 – 7 \lambda + 11$$

Factoring the characteristic polynomial, we get:

$$(\lambda – 2) (\lambda – 5) = 0$$

Therefore, the eigenvalues of the matrix are $\lambda = 2$ and $\lambda = 5$.

To explain each option in brief:

• Option A: $-1$ and $1$. These are not eigenvalues of the matrix, since the characteristic polynomial does not have roots at $-1$ and $1$.
• Option B: $1$ and $6$. These are not eigenvalues of the matrix, since the characteristic polynomial does not have roots at $1$ and $6$.
• Option C: $2$ and $5$. These are the eigenvalues of the matrix, since the characteristic polynomial has roots at $2$ and $5$.
• Option D: $4$ and $-1$. These are not eigenvalues of the matrix, since the characteristic polynomial does not have roots at $4$ and $-1$.