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Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
Let $A$ be a $4\times 4$ matrix with real entries such that $-1,1,2,-2$ are its eigenvalues. If $B=A^{4}-5A^{2}+5I$, where $I$ denotes the $4\times 4$ identity matrix, then which of the following statements are correct?
- $\det (A+B)=0$
- $\det B=1$
- $\text{trace}(A-B)=0$.
- $\text{trace}(A+B)=4$.
NOTE: There may one or more options correct.
I know that trace of matrix means sum of eigenvalues of matrix and determinant means product of eigenvalues. But i dont know how to apply these things in this question?
Please help me out and explain the method.