Question

Let `M` be a `3xx3` matrix satisfying `M[0 1 0]=M[1-1 0]=[1 1-1],a n dM[1 1 1]=[0 0 12]` Then the sum of the diagonal entries of `M` is _________.

Answer 1

Correct Answer - 9
Let `M =|{:(a_(1), a_(2), a_(3)), (b_(1), b_(2), b_(3)), (c_(1), c_(2), c_(3)):}|`
`therefore M [(0), (1),(0)] rArr M [(1), (-1), (0)] = [(1), (1), (-1)]`
`M * [(1), (1), (1)] = [(0), (0), (12)]`
`rArr [(a_(2)), (b_(2)), (c_(2))] = [(-1), (2), (3)] * [(a_(1)-a_(2)), (b_(1)-b_(2)), (c_(1), c_(2))] = [(1), (1), (-1)], [(a_(1), + a_(2) + a_(3)), (b_(1), + b_(2) + b_(3)), (c_(1), + c_(2) + c_(3))] = [(0), (0), (12)]`
`rArr a_(2) = -1, b_(2) = 2, c_(2) = 3, a_(1)-a_(2) =1, b_(1)-b_(2) = 1, c_(1)-c_(2) =-1`
`rArr a_(1) + a_(2) +a_(3) =0, b_(1) +b_(2) + b_(3) =0, c_(1) + c_(2) + c_(3) = 12`
`therefore a_(1) =0, b_(2) = 2, c_(3) =7`
`rArr` Sum of diagonal elements `=0+2+7 =9`