are 1 and 4
are -1 and 2
are 0 and 5
cannot be determined
Answer is Right!
Answer is Wrong!
The correct answer is A.
The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a matrix $A$ is given by $$p(x) = |xI – A|.$$
In this case, we have $$p(x) = |xI – \left[ {\begin{array}{*{20}{c}} 4&{ – 2} \ { – 2}&1 \end{array}} \right]| = |x – 4| |x – 1| = (x – 4)(x – 1).$$
Therefore, the eigenvalues of the matrix are $4$ and $1$.
Option B is incorrect because the eigenvalues are not $-1$ and $2$. Option C is incorrect because the eigenvalues are not $0$ and $5$. Option D is incorrect because the eigenvalues can be determined.