Question

Let \( A \) be a \( 3 \times 3 \) real matrix having eigenvalues \( 1, 0, \) and \( -1 \). If \( B = A^2 + 2A + I_3 \), where \( I_3 \) is the \( 3 \times 3 \) identity matrix, then which one of the following statements is true?

• A. \( B^3 - 5B^2 + 4B = 0 \)
• B. \( B^3 - 5B^2 - 4B = 0 \)
• C. \( B^3 + 5B^2 - 4B = 0 \)
• D. \( B^3 + 5B^2 + 4B = 0 \)

Hint:

- The matrix equation can be solved using the properties of eigenvalues.
- The characteristic polynomial helps us express the equation satisfied by a matrix.
- Eigenvalue computations often simplify matrix problem-solving.

Answer 1

The Correct Option is A

1) Understanding the given condition:
We are given that \( A \) is a \( 3 \times 3 \) matrix with eigenvalues \( 1, 0, -1 \). The matrix \( B \) is defined as: \[ B = A^2 + 2A + I_3 \] The task is to determine which of the following four statements about \( B \) is true.
2) Eigenvalues of \( B \):
Since \( B = A^2 + 2A + I_3 \), we can calculate the eigenvalues of \( B \) based on the eigenvalues of \( A \). Let the eigenvalues of \( A \) be \( \lambda \). The corresponding eigenvalue of \( B \) is computed as: \[ \mu = \lambda^2 + 2\lambda + 1 \] Thus, the eigenvalues of \( B \) are:
- For \( \lambda = 1 \): \( \mu = 1^2 + 2(1) + 1 = 4 \)
- For \( \lambda = 0 \): \( \mu = 0^2 + 2(0) + 1 = 1 \)
- For \( \lambda = -1 \): \( \mu = (-1)^2 + 2(-1) + 1 = 0 \)
Thus, the eigenvalues of \( B \) are \( 4, 1, 0 \).
3) Characteristic polynomial of \( B \):
The characteristic polynomial of \( B \) is obtained from its eigenvalues: \[ (B - 4)(B - 1)(B) = 0 \] Expanding this, we get the equation: \[ B^3 - 5B^2 + 4B = 0 \]
4) Conclusion:
From the derived equation, it is evident that the correct statement is (A): \( B^3 - 5B^2 + 4B = 0 \).