Matrices & Determinants
Properties of Determinants
Grade 12
Question:
<p>The digits <span>A</span>, <span>B</span>, <span>C</span> are such that the three digit numbers <span>A88</span>, <span>6B8</span>, <span>86C</span> are divisible by 72. The determinant <span>\begin{vmatrix} A & 6 & 8 \\ 8 & B & 6 \\ 8 & 8 & C \end{vmatrix}</span> is divisible by</p>
<p>(a) 72</p>
<p>(b) 144</p>
<p>(c) 288</p>
<p>(d) 216</p>
Step-by-Step Solution
Key Concept: Use divisibility conditions to find the digits, then compute the determinant and check which option divides it.
<p><strong>Step 1:</strong> Since <span>A88</span>, <span>6B8</span>, <span>86C</span> are divisible by 72, they must also be divisible by 9.</p><p><strong>Step 2:</strong> For divisibility by 9: <span>A + 8 + 8</span>, <span>6 + B + 8</span>, <span>8 + 6 + C</span> must be divisible by 9.</p><p><strong>Step 3:</strong> This gives us <span>A = 2</span>, <span>B = 4</span>, <span>C = 4</span>.</p><p><strong>Step 4:</strong> Calculate the determinant: <span>\begin{vmatrix} 2 & 6 & 8 \\ 8 & 4 & 6 \\ 8 & 8 & 4 \end{vmatrix} = 2(16 - 48) - 6(32 - 48) + 8(64 - 32) = 2(-32) - 6(-16) + 8(32) = -64 + 96 + 256 = 288</span></p><p><strong>Step 5:</strong> The determinant equals 288, which is divisible by 72, 144, and 288. The answer is <strong>288</strong>.</p>
Correct Answer: C