The total number of 3×3 matrices A having entries from the set {0,1,2,3} such that the sum of  all the diagonal entries of AAT is 9 , is equal to

A=abcdefghi⇒AT=adgbehcfiAAT=a2+b2+c2ad+be+cf  ag+bh+ci da+eb+fcd2+e2+f2dg+eh+figa+hb+icgd+he+ifg2+h2+i2 Given, a2+b2+c2+d2+e2+f2+g2+h2+i2=9 and a,b,c,d,e,f,g,h,i∈{0,1,2,3}Case(i)  One of them is 3, remaining are 0s is 9 ways Case (ii) 2 of them are 2 s , one of them is 1 and remaining 0 s is 9c2·7C1=252 ways  Case (iii) One of them is 2 , five of them are 1 s and remaining 0 s is 9c1·8c5=504 Ways Case iv All are 1s in one way Total: 9+252+504+1=767 ways