The eigen values of matrix \[\left[ {\begin{array}{*{20}{c}} 9&5 \\ 5&8 \end{array}} \right]\] are A. -2.42 and 6.86 B. 3.48 and 13.53 C. 4.70 and 6.86 D. 6.86 and 9.50

The correct answer is $\boxed{\text{A.}}$ -2.42 and 6.86.

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 $\left[ {\begin{array}{*{20}{c}} 9&5 \ 5&8 \end{array}} \right]$ is $p(x) = x^2 – 17x + 72$.

Solving for the roots of $p(x)$, we find that the eigenvalues are $\lambda = -2.42$ and $\lambda = 6.86$.

Here is a brief explanation of each option:

• Option A: $-2.42$ and $6.86$ are the eigenvalues of the matrix $\left[ {\begin{array}{*{20}{c}} 9&5 \ 5&8 \end{array}} \right]$.
• Option B: $3.48$ and $13.53$ are not eigenvalues of the matrix $\left[ {\begin{array}{*{20}{c}} 9&5 \ 5&8 \end{array}} \right]$.
• Option C: $4.70$ and $6.86$ are not eigenvalues of the matrix $\left[ {\begin{array}{*{20}{c}} 9&5 \ 5&8 \end{array}} \right]$.
• Option D: $6.86$ and $9.50$ are not eigenvalues of the matrix $\left[ {\begin{array}{*{20}{c}} 9&5 \ 5&8 \end{array}} \right]$.