The Eigen values of the matrix \[\left[ {\text{P}} \right] = \left[ {\begin{array}{*{20}{c}} 4&5 \\ 2&{ – 5} \end{array}} \right]\] are A. -7 and 8 B. -6 and 5 C. 3 and 4 D. 1 and 2

The correct answer is $\boxed{\text{A}}$.

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a matrix $A$ is given by

$$p(x) = |xI – A|$$

where $I$ is the identity matrix.

In this case, the characteristic polynomial of $P$ is

$$p(x) = |xI – P| = \left| \begin{array}{cc} x – 4 & -5 \\ 2 & x + 5 \end{array} \right| = x^2 – x – 30$$

To find the eigenvalues, we need to solve the equation $p(x) = 0$. This gives us the eigenvalues $x = -7$ and $x = 8$.

Therefore, the eigenvalues of the matrix $P$ are $\boxed{-7}$ and $\boxed{8}$.

Here is a brief explanation of each option:

• Option A: The eigenvalues of $P$ are $-7$ and $8$. This is the correct answer.
• Option B: The eigenvalues of $P$ are $-6$ and $5$. This is not the correct answer, as the eigenvalues of $P$ are not $-6$ and $5$.
• Option C: The eigenvalues of $P$ are $3$ and $4$. This is not the correct answer, as the eigenvalues of $P$ are not $3$ and $4$.
• Option D: The eigenvalues of $P$ are $1$ and $2$. This is not the correct answer, as the eigenvalues of $P$ are not $1$ and $2$.