The number of 3 2 matrices A, which can be formed using the elements of the set -2,-1,0,1,2 such that the sum of all the diagonal elements of A ^ T A is 5, is

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: a₁₁^2 + a₂₁^2 + a₃₁^2 + a₁₂^2 + a₂₂^2 + a₃₂^2 = 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: