How many matrices with entries from are there, for which the sum of the diagonal entries of is 5?
- A.198
- B.162
- C.126
- D.135
Show answer and solution
Correct option: A
Solution
The diagonal entries of MT M sum to the sum of squares of ALL 9 entries of M. So we need the number of 3×3 matrices with entries from {0,1,2} where the sum of squares of entries equals 5.
Each entry contributes 0² = 0, 1² = 1, or 2² = 4. We need combinations of nine such values summing to 5.
Case 1: One entry is 2 (contributes 4) and one entry is 1 (contributes 1), rest are 0.
Number of ways = 9 × 8 = 72 (choose position for 2, then position for 1).
Case 2: Zero entries are 2, and five entries are 1 (each contributing 1), rest are 0.
Number of ways = C(9,5) = 126.
Total = 72 + 126 = 198.
Each entry contributes 0² = 0, 1² = 1, or 2² = 4. We need combinations of nine such values summing to 5.
Case 1: One entry is 2 (contributes 4) and one entry is 1 (contributes 1), rest are 0.
Number of ways = 9 × 8 = 72 (choose position for 2, then position for 1).
Case 2: Zero entries are 2, and five entries are 1 (each contributing 1), rest are 0.
Number of ways = C(9,5) = 126.
Total = 72 + 126 = 198.
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