Let M denote the set of all real matrices of order 3 3 and let S = -3,-2,-1,1,2 . Let aligned & S ₁= A = [a_ ij ] M : A = A ^ T and a_ ij ~S , i , j , & S ₂= A = [a_ ij ] M : A =- A ^ T and a_ ij ~S , i , j , & S ₃= A = [a_ ij ] M : a₁₁+a₂₂+a₃₃=0 and a_ ij ~S , i , j . aligned If n ( ~S ₁ ₂ US ₃ )=125 , then equals _______

Step-by-step solution

[ array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ] No. of elements in S ₁: A = A ^ T 5^3 5^3 No. of elements in A=-A^T 0 since no. zero in 5 No. of elements in S ₃ . array c a₁₁+a₂₂+a₃₃=0 (1,2,-3) 31 or (1,1,-2) 3 or (-1,-1,2) 3 array 12 5^6 aligned & n ( S ₁ ~S ₃ )=12 5^3 & n ( S ₁ ~S ₂ ~S ₃ )=5^6(1+12)-12 5^3 & 5^3 [13 5^3-12 ]=125 & =1613 aligned