Given that a $3 \times 3$ matrix $B$ has eigenvalues $0$, $1$, and $2$, we can analyze the options one by one:

a. The rank of $B$:

The rank of a matrix is the dimension of its column space (or row space), which is the same as the number of non-zero eigenvalues.

• Since $B$ has eigenvalues $0$, $1$, and $2$, the non-zero eigenvalues are $1$ and $2$.
• Thus, the rank of $B$ is $2$ (since the eigenvalue $0$ contributes nothing to the rank).

Conclusion: The rank of $B$ can be determined. (Correct choice)

b. The determinant of $B^T B$:

The determinant of a matrix is the product of its eigenvalues.

• $B^T B$ is a symmetric matrix, and its eigenvalues are the squares of the singular values of $B$, which are the absolute values of the eigenvalues of $B$.
• The eigenvalues of $B$ are $0$, $1$, and $2$, so the eigenvalues of $B^T B$ are $0^2 = 0$, $1^2 = 1$, and $2^2 = 4$.
• The determinant of $B^T B$ is the product of these eigenvalues: $0 \times 1 \times 4 = 0$.

Conclusion: The determinant of $B^T B$ can be determined. (Correct choice)

c. The eigenvalues of $B^T B$:

• As discussed in part (b), the eigenvalues of $B^T B$ are the squares of the absolute values of the eigenvalues of $B$.
• Given that the eigenvalues of $B$ are $0$, $1$, and $2$, the eigenvalues of $B^T B$ are $0$, $1$, and $4$.

Conclusion: The eigenvalues of $B^T B$ can be determined. (Correct choice)

d. The eigenvalues of $(B + I)^{-1}$:

To determine the eigenvalues of $(B + I)^{-1}$, we can use the fact that if $\lambda$ is an eigenvalue of $B$, then $\frac{1}{\lambda + 1}$ will be an eigenvalue of $(B + I)^{-1}$.

• The eigenvalues of $B$ are $0$, $1$, and $2$.
• Thus, the eigenvalues of $B + I$ will be $0 + 1 = 1$, $1 + 1 = 2$, and $2 + 1 = 3$.
• The eigenvalues of $(B + I)^{-1}$ will be $\frac{1}{1} = 1$, $\frac{1}{2} = 0.5$, and $\frac{1}{3} \approx 0.333$.

Conclusion: The eigenvalues of $(B + I)^{-1}$ can be determined. (Correct choice)

Final Conclusion:

All four choices $a$, $b$, $c$, and $d$ can be determined from the given information about the eigenvalues of $B$.

Would you like further details on any of these points? Here are five related questions:

1. How is the rank of a matrix related to its eigenvalues?
2. What does the determinant of a matrix tell us about its eigenvalues?
3. How are the eigenvalues of $B$ related to the eigenvalues of $B^T B$?
4. Can the eigenvalues of a matrix be negative, and how does that affect the eigenvalues of $B^T B$?
5. What properties of a matrix are preserved under similarity transformations?

Tip: Understanding how eigenvalues transform under matrix operations can help solve a wide range of problems in linear algebra.