4, 9
6, -8
4, 8
-6, 8
Answer is Right!
Answer is Wrong!
The correct answer is $\boxed{\text{A}. 4, 9}$.
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 $\left[ {\begin{array}{*{20}{c}} {10}&{ – 4} \ {18}&{ – 12} \end{array}} \right]$ is:
$$p(x) = x^2 – 2x – 36$$
The discriminant of the characteristic polynomial is:
$$b^2 – 4ac = 4^2 – 4 \cdot 1 \cdot (-36) = 192$$
The eigenvalues of the matrix are:
$$\lambda = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} = \frac{-2 \pm \sqrt{192}}{2 \cdot 1} = 4 \pm 8 = \boxed{4, 9}$$