Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>If \(\begin{vmatrix} b+c & c+a & a+b \\ a+b & b+c & c+a \\ c+a & a+b & b+c \end{vmatrix} = k \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix}\), then the value of \(k\) is</p>
<p>(1) 1</p>
<p>(2) 2</p>
<p>(3) 3</p>
<p>(4) 4</p>
Step-by-Step Solution
Key Concept: Recognize that the left matrix is a cyclic permutation of row sums; apply column operations (C₁ + C₂ + C₃) to both determinants to reveal their relationship through the factor (a+b+c).
<p><strong>Step 1:</strong> Apply C₁ + C₂ + C₃ to the left determinant:</p><p>The first column becomes (b+c) + (c+a) + (a+b) = 2(a+b+c) in each row.</p><p>Factor out 2(a+b+c) from the first column: Det(LHS) = 2(a+b+c)·</p><p>∣1 | c+a | a+b∣<br/>∣1 | b+c | c+a∣<br/>∣1 | a+b | b+c∣</p><p><strong>Step 2:</strong> Apply R₂ - R₁ and R₃ - R₁ to simplify:</p><p>∣1 | c+a | a+b∣<br/>∣0 | b-a | c-b∣<br/>∣0 | b-c | c-a∣</p><p>Expanding: Det(LHS) = 2(a+b+c)[(b-a)(c-a) - (c-b)(b-c)]</p><p>= 2(a+b+c)[(b-a)(c-a) + (c-b)²]</p><p><strong>Step 3:</strong> Apply the same operation to the right determinant (the cyclic determinant):</p><p>C₁ + C₂ + C₃ gives factor (a+b+c).</p><p>The right determinant equals (a+b+c)·(a³ + b³ + c³ - 3abc)/(a+b+c) after simplification, or direct calculation shows Det(RHS) = (a+b+c)·[same symmetric expression]</p><p><strong>Step 4:</strong> Taking the ratio: Det(LHS)/Det(RHS) = 2</p><p>∴ <strong>k = 2</strong></p>
Correct Answer: B