Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>Let \[D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}\] and \[D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}\] then the value of \(\dfrac{D_1}{D_2}\), where \(b \neq 0\) and \(ad \neq bc\), is ________.</p>
Step-by-Step Solution
Key Concept: Use column operations to simplify determinants: subtract one column from another to reveal proportional rows or columns, which collapse the determinant to a factored form involving a common factor.
<p><strong>Step 1: Simplify D₁ using column operations</strong></p><p>For D₁, perform C₃ → C₃ - C₁ - C₂:</p><p>D₁ = |a b (a+b)-a-b| = |a b 0|</p><p> |c d (c+d)-c-d| |c d 0|</p><p> |a b (a-b)-a-b| |a b -2b|</p><p>Expanding along C₃: D₁ = -2b · |a b| = -2b(ad - bc)</p><p> |c d|</p><p><strong>Step 2: Simplify D₂ using column operations</strong></p><p>For D₂, perform C₃ → C₃ - C₁ - C₂:</p><p>D₂ = |a c (a+c)-a-c| = |a c 0|</p><p> |b d (b+d)-b-d| |b d 0|</p><p> |a c (a+b+c)-a-c| |a c b|</p><p>Expanding along C₃: D₂ = b · |a c| = b(ad - bc)</p><p> |b d|</p><p><strong>Step 3: Calculate the ratio</strong></p><p>D₁/D₂ = [-2b(ad - bc)] / [b(ad - bc)] = -2b/b = <strong>-2</strong></p><p>∴ Answer: <strong>-2</strong></p>
Correct Answer: -2