Eigenvectors and Orthogonal matrix

Question

"Knowing that a matrix $B\in \Bbb{R^{3\times 3}}$ has eigenvalues $0$, $1$ e $2$. This information is enough to calculate three of the next items:

$(i)$ Rank of $B$;

$(ii)$ Determinant of $B^T B$;

$(iii)$ Eigenvalues of $B^T B$;

$(iv)$ Eigenvalues of $(B+I)^{-1}$.

What is the item that needs additional informations to be calculated? Calculate the other three."

The $(i)$ and $(iv)$ I know how to calculate but I don't know about the others .Someone posted Finding $(B^2+I)^{-1}$ using eigenvalues and size. that is almost equal, however the answer is not complete.

@Berci When I wrote that https://math.stackexchange.com/questions/2059736/finding-b2i-1-using-eigenvalues-and-size is not complete I meant that only $(iv)$ is commented. Can you show me how to get $(ii)$? — Rebeca Lie Yatsuzuka Silva, Mar 08 '20 at 16:27

score 0 · Accepted Answer · answered Mar 08 '20 at 17:09

0

Hint: for (ii), we have $\det(AB)=\det A\cdot \det B$ for any square matrices, apply it with $A=B^T$ and observe $\det B=0$.

(By the way, $\det(B^T) =\det B$, so $\det(B^TB) =(\det B)^2$ also holds for any square matrix $B$.)

answered Mar 08 '20 at 17:09

Berci

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Eigenvectors and Orthogonal matrix

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