<p>If \(D_1\) and \(D_2\) are two \(3 \times 3\) diagonal matrices, then which of the following is true?</p>
<p>\(D_1 D_2\) is a diagonal matrix</p>
<p>\(D_1 D_2 = D_2 D_1\)</p>
<p>\(D_1^2 + D_2^2\) is a diagonal matrix</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: Diagonal matrices commute under multiplication (D₁D₂ = D₂D₁), and their product is also diagonal. Key properties: diagonal matrices form a commutative subring, products preserve the diagonal structure, and they're simultaneously diagonalizable.
<p><strong>Step 1: Product of Diagonal Matrices</strong></p><p>If D₁ = diag(a₁, a₂, a₃) and D₂ = diag(b₁, b₂, b₃), then:</p><p>D₁D₂ = diag(a₁b₁, a₂b₂, a₃b₃) ✓ (Product is diagonal)</p><p><strong>Step 2: Commutativity Check</strong></p><p>D₁D₂ = diag(a₁b₁, a₂b₂, a₃b₃) = diag(b₁a₁, b₂a₂, b₃a₃) = D₂D₁ ✓ (Diagonal matrices commute)</p><p><strong>Step 3: Determinant Properties</strong></p><p>det(D₁D₂) = (a₁b₁)(a₂b₂)(a₃b₃) = (a₁a₂a₃)(b₁b₂b₃) = det(D₁)·det(D₂) ✓</p><p><strong>Step 4: Trace Properties</strong></p><p>tr(D₁D₂) = a₁b₁ + a₂b₂ + a₃b₃ = tr(D₂D₁) ✓</p><p><strong>Step 5: Invertibility (when applicable)</strong></p><p>If D₁ and D₂ are both invertible, D₁D₂ is invertible and (D₁D₂)⁻¹ = D₂⁻¹D₁⁻¹ = D₁⁻¹D₂⁻¹ ✓</p><p>∴ True statements: D₁D₂ is diagonal, D₁D₂ = D₂D₁, det(D₁D₂) = det(D₁)det(D₂), and tr(D₁D₂) = tr(D₂D₁)</p>
Correct Answer: A,B,C