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Let M be a 3xx3 matrix satisfying M[(0...

Let M be a `3xx3` matrix satisfying
`M[(0),(1),(0)]=[(-1),(2),(3)], M[(1),(-1),(0)]=[(1),(1),(-1)]`, and `M[(1),(1),(1)]=[(0),(0),(12)]`
Then the sum of the diagonal entries of M is ____.

Text Solution

Verified by Experts

The correct Answer is:
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`
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