Question

Let \( A \) be a \( 3 \times 3 \) real matrix. Suppose that 1 and 2 are characteristic roots of \( A \), and 12 is a characteristic root of \( A + A^2 \). Then which of the following statements is/are correct?

• A. \(\det(A) \neq 0\)
• B. \(\det(A + A^2) \neq 0\)
• C. \(\det(A) = 0\)
• D. trace of \((A + A^2)\) is 20

Answer 1

The Correct Option is A, B, D

1. Characteristic Roots of \( A \): - Since \( A \) is a \( 3 \times 3 \) matrix and has 1 and 2 as characteristic roots, the third root, say \( \lambda_3 \), must satisfy: \[ \det(A) = 1 \cdot 2 \cdot \lambda_3. \] 2. Condition for \( A + A^2 \): - The characteristic roots of \( A + A^2 \) are \( \lambda + \lambda^2 \), where \( \lambda \) is a characteristic root of \( A \). For \( \lambda = 1 \) and \( \lambda = 2 \): \[ \text{For } \lambda = 1, \quad 1 + 1^2 = 2. \] \[ \text{For } \lambda = 2, \quad 2 + 2^2 = 6. \] \[ \text{For } \lambda_3, \quad \lambda_3 + \lambda_3^2 = 12 \implies \lambda_3 = 3. \] 3. Determinant of \( A \): - Using \( \lambda_3 = 3 \), the determinant of \( A \) is: \[ \det(A) = 1 \cdot 2 \cdot 3 = 6 \neq 0. \] 4. Trace of \( A + A^2 \): - The trace of \( A + A^2 \) is the sum of its characteristic roots: \[ \text{Trace}(A + A^2) = (1 + 1^2) + (2 + 2^2) + (3 + 3^2) = 2 + 6 + 12 = 20. \] However, this is not relevant to the determinant of \( A \). 5. Correct Statement: - From the above, \( \det(A) \neq 0 \).