Matrices & Determinants
Determinant Properties
Grade 12
Question:
<p>If <i>a</i>₁<i>b</i>₁<i>c</i>₁, <i>a</i>₂<i>b</i>₂<i>c</i>₂ and <i>a</i>₃<i>b</i>₃<i>c</i>₃ are three digit even natural numbers and \(\Delta = \begin{vmatrix} c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3 \end{vmatrix}\), then \(\Delta\) is</p>
<p>(a) divisible by 2 but not necessarily by 4</p>
<p>(b) divisible by 4 but not necessarily by 8</p>
<p>(c) divisible by 8</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: If all elements of a determinant are even, factor 2 from each row to determine divisibility.
<p>Each digit (<i>a</i>ᵢ, <i>b</i>ᵢ, <i>c</i>ᵢ) of even three-digit numbers is even. Since all elements are even, we can factor out 2 from each row. With 3 rows: $\Delta = 2^3 \times \Delta' = 8\Delta'$, so $\Delta$ is divisible by 8.</p>
Correct Answer: C