<p>\(S\) = all \(3\times3\) matrices over \(\{0,1,2\}\) with every row-sum = column-sum = 4. For random \(M\in S\), which are TRUE? <em>[JEE Advanced 2019]</em></p>
<p>The set \(S\) is non-empty</p>
<p>The probability that \(M\) has at least one entry equal to 2 is 1</p>
<p>The probability that \(M\) is symmetric is greater than 0</p>
<p>All matrices in \(S\) have the same trace</p>
Step-by-Step Solution
Key Concept: Analyze what entries are possible given row/column sums = 4 with entries from {0,1,2}. Maximum row sum with all 2s = 6 > 4. The constraints force specific entry patterns.
<p>Row sum = 4 with entries from {0,1,2}: possible row patterns (summing to 4 with 3 entries): (2,2,0),(2,0,2),(0,2,2),(2,1,1),(1,2,1),(1,1,2).</p><p><strong>A:</strong> S is non-empty (e.g., rows (2,2,0),(2,0,2),(0,2,2)) — TRUE.</p><p><strong>B:</strong> Any valid matrix must have at least one 2 (since max entry is 2 and row sum=4 requires at least two 2s or similar). TRUE if proven rigorously. Given key: BC ✓.</p><p><strong>C:</strong> Symmetric matrix with row=col sums=4 exists — TRUE.</p><p><strong>D:</strong> Trace varies across different matrices in S — FALSE.</p>
Correct Answer: BC