Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>The value of the determinant \[\begin{vmatrix} (a_1-b_1)^2 & (a_1-b_2)^2 & (a_1-b_3)^2 & (a_1-b_4)^2 \\ (a_2-b_1)^2 & (a_2-b_2)^2 & (a_2-b_3)^2 & (a_2-b_4)^2 \\ (a_3-b_1)^2 & (a_3-b_2)^2 & (a_3-b_3)^2 & (a_3-b_4)^2 \\ (a_4-b_1)^2 & (a_4-b_2)^2 & (a_4-b_3)^2 & (a_4-b_4)^2 \end{vmatrix}\] is</p>
<p>(1) dependant on \(a_i\), \(i = 1, 2, 3, 4\)</p>
<p>(2) dependant on \(b_i\), \(i = 1, 2, 3, 4\)</p>
<p>(3) dependant on \(a_i\), \(b_i\), \(i = 1, 2, 3, 4\)</p>
<p>(4) 0</p>
Step-by-Step Solution
Key Concept: This 4×4 matrix has rank at most 2 because each entry can be expressed as (aᵢ - bⱼ)² = aᵢ² - 2aᵢbⱼ + bⱼ², making it a sum of at most 3 simpler matrices. A matrix of rank < n has determinant 0.
<p><strong>Step 1:</strong> Recognize the structure. Each element can be written as:</p><p>(aᵢ - bⱼ)² = aᵢ² - 2aᵢbⱼ + bⱼ²</p><p><strong>Step 2:</strong> Express the matrix as a sum of three simpler matrices:</p><p>M = A + B + C where:</p><p>• A has entries aᵢ² (constant in each row)</p><p>• B has entries -2aᵢbⱼ (rank 1: outer product of vectors)</p><p>• C has entries bⱼ² (constant in each column)</p><p><strong>Step 3:</strong> The rank of M is at most 3 (actually at most 2 due to the constraints). Since M is 4×4, if rank(M) < 4, then det(M) = 0.</p><p><strong>Step 4:</strong> More directly: the 4×4 matrix M can be expressed as M = uvᵀ + vwᵀ + wuᵀ type combinations where each component is at most rank 1. The sum of three rank-1 matrices has rank ≤ 3, so for a 4×4 matrix, det(M) = 0.</p><p>∴ <strong>Answer: D (which is 0)</strong></p>
Correct Answer: D