A 3 × 3 matrix B is known to have eigenvalues 0, 1, 2. This information is enough to find three of
these:
a. the rank of B,
b. the determinant of BT B,
c. the eigenvalues of BT B,
d. the eigenvalues of (B + I)^(-1)
Mark your choices. Explain/justify

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.

Explore how eigenvalues of a 3x3 matrix B (with eigenvalues 0, 1, and 2) help determine its rank, the determinant of B^T B, the eigenvalues of B^T B, and the eigenvalues of (B + I)^(-1). Learn key linear algebra concepts and formulas used in matrix analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation