CBSE Class 12-science Answered
Sir pl solve questions no 15
Asked by shobhit | 29 Mar, 2019, 10:56: PM
Expert Answer
Let matrix A and matrix B are non-singular matrices. Hence A-1 and B-1 exsist and det(A) ≠ 0 and det(B) ≠ 0 .
we have, A-1AB = B ......................(1)
AB B-1 = A ..........................(2)
since we are given that AB is null matrix, we get from (1) and (2), A = B = 0,
which is contradictory to our assumption that a nd B are non-singular .
Hence statement-(A) : det(A)≠0 means B must be null matrix is true
Similarly statement-(B) : det(B)≠0 means A must be null matrix is true
statement-(C) If none of A and B are null matrices, then atleast one of the matrix should be singular. This is true
If none of A and B are null matrices, but the product AB is null matrix, then det(AB) = 0
we have det(AB) = det(A)×det(B) .......................(3)
in eqn.(3) LHS is zero, hence atleast one determinant in RHS should be zero.
statement-(D) :- If neither det(A) nor det(B) is zero, then the given statement in the question is not possible. This is true
If neither det(A) nor det(B) is zero, then RHS of eqn.(3) is non-zero. But LHS is zero because AB is null matrix.
Hence atleast one of the determinent is zero
Answered by Thiyagarajan K | 30 Mar, 2019, 05:05: PM
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