<p>Which of the following is (are) NOT the square of a \(3 \times 3\) matrix with real entries?</p>
<p>\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: A matrix B is a perfect square of a real matrix if there exists a real matrix A such that A² = B. Use eigenvalue analysis: if B = A², then eigenvalues of B must be perfect squares of real numbers, and complex eigenvalues must appear in conjugate pairs with matching multiplicities to guarantee real A exists.
<p><strong>Key Principle:</strong> If B = A² for real matrix A, then:</p><p>(1) If λ is a real eigenvalue of B, then √λ must be real (allowing ±√λ in A)</p><p>(2) Complex eigenvalues of B must appear in conjugate pairs, and when we take square roots, they must generate real entries in A</p><p><strong>Critical Test:</strong> If B has an eigenvalue that is a negative real number that's not a perfect square, or complex eigenvalues whose square roots cannot be arranged to form a real matrix, then B ≠ A² for any real A.</p><p><strong>Eliminating Options:</strong></p><p>• Matrices with eigenvalue -1 or other negative reals cannot be squares of real matrices (would require ±i in the matrix A, breaking reality)</p><p>• Complex conjugate pair eigenvalues (a±bi) of B require √(a+bi) to generate real entries—this fails for most pairs</p><p><strong>Method:</strong> Calculate characteristic polynomial → find eigenvalues → verify if each eigenvalue λ can be written as μ² for some real μ, AND that complex pairs can simultaneously yield real A. If either fails, the matrix is NOT a perfect square of a real matrix.</p><p>∴ Answer: AC (the two matrices that violate these conditions)</p>
Correct Answer: AC