JEE Main Mathematics Question (2026) — Solution

Step-by-step solution

For a 3×2 matrix A, the diagonal sum of A^T A equals the sum of squares of all 6 elements of A. We need: a_ 11 ^2 + a_ 21 ^2 + a_ 31 ^2 + a_ 12 ^2 + a_ 22 ^2 + a_ 32 ^2 = 5 where each element is from \ -2, -1, 0, 1, 2\ . Possible squares: 0^2=0, ( 1)^2=1, ( 2)^2=4. Case 1: Distribution 4+1+0+0+0+0 (one ±2, one ±1, four 0's). Arrangements: 6 1 5 1 = 30. Sign choices: 2 2 = 4. Total: 30 4 = 120. Case 2: Distribution 1+1+1+1+1+0 (five ±1, one 0). Arrangements: 6 1 = 6 ways to place the 0. Sign choices for five ±1: 2^5 = 32. Total: 6 32 = 192. Total matrices: 120 + 192 = 312