Let A = a 0 3 0 b 5 - 1 3 0 c , where a , b , c are positive integers. If t r A = 7 , then the greatest value of A is (where t r A denotes the trace of matrix A i.e. the sum of principal diagonal elements of matrix A )

Step-by-step solution

t r A = a + b + c = 7 Ordered triplets of a , b , c are 1,1 , 5 , 1,2 , 4 , 1,3 , 3 , 2 ,1 , 4 , 2 ,2 , 3 , 2 ,3 , 2 , 2 ,4 , 1 , 3 ,1 , 3 , 3 ,2 , 2 , 3 ,3 , 1 , 4 ,1 , 2 , 4 ,2 , 1 , 5 ,1 , 1 A = a 0 3 0 b 5 - 1 3 0 c = a b c + 3 b 3 = a b c + b A is greatest for 2,3 , 2 and equal to 15