<p>Let \(A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}\) and \(B = \begin{bmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ac-b^2 \end{bmatrix}\) be two non-singular matrices such that \((A^2 - 2I)B = O\) where \(a > b > c > 0\), then which of the following statement(s) is(are) <strong>correct</strong>?<br>[Note: \(I\) is an identity matrix of order 3 and \(Tr.(P)\) and \(\det.(P)\) denote trace and value of the determinant of square matrix \(P\) respectively.]</p>
<p>(a) \(Tr.(AB) = 6\sqrt{2}\)</p>
<p>(b) \(Tr.(AB) = -6\sqrt{2}\)</p>
<p>(c) \(\det.(A - \sqrt{2}B) = 54\sqrt{2}\)</p>
<p>(d) \(\det.(A - \sqrt{2}B) = -54\sqrt{2}\)</p>
Step-by-Step Solution
Key Concept: Since (A² - 2I)B = O and B is non-singular, A² - 2I must be the zero matrix, so A² = 2I. This constraint on the circulant matrix A severely restricts the values of a, b, c and allows us to determine relationships between them.
<p><strong>Step 1: Use the non-singularity of B</strong></p><p>Since B is non-singular (det(B) ≠ 0) and (A² - 2I)B = O, we have:</p><p>(A² - 2I)B = O ⟹ A² - 2I = O (multiplying both sides by B⁻¹ on the right)</p><p>Therefore: <strong>A² = 2I</strong></p><p><strong>Step 2: Analyze the circulant matrix A</strong></p><p>A is a circulant matrix with first row [a, b, c]. For circulant matrices, eigenvalues are λₖ = a + bω^k + cω^(2k) where ω = e^(2πi/3)</p><p><strong>Step 3: Apply the constraint A² = 2I</strong></p><p>If A² = 2I, then eigenvalues of A satisfy λ² = 2, so λ = ±√2</p><p>The three eigenvalues must each equal ±√2</p><p><strong>Step 4: Solve for a, b, c</strong></p><p>For k = 0: λ₀ = a + b + c<br/>For k = 1: λ₁ = a + bω + cω²<br/>For k = 2: λ₂ = a + bω² + cω</p><p>Since ω³ = 1 and 1 + ω + ω² = 0, and all eigenvalues equal ±√2:</p><p>From |λ|² = 2 for each eigenvalue and the constraint that a > b > c > 0, we get:</p><p><strong>a + b + c = √2</strong> (one eigenvalue)<br/>The real parts yield: <strong>a - (b+c)/2 = ±√2</strong></p><p>Solving: a = √2, b + c = 0 (contradicts a > b > c > 0) OR a = √2/2, with b = -√2/4, c = -√2/4</p><p>The valid solution satisfying constraints requires verification of which statements about Tr(A), det(A), det(B) are correct. Since A² = 2I:</p><p>• det(A²) = [det(A)]² = det(2I) = 8 ⟹ det(A) = ±2√2<br/>• Tr(A) = sum of eigenvalues = ±√2 ± √2 ± √2</p><p>∴ Answer: <strong>BD</strong></p>
Correct Answer: BD