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Question

Let A=[abcd] and B=[pq][00]. If AB=B and a+d=2, then the value of adbc is

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Solution

AB=B
[abcd][pq]=[pq]

[ap+bqcp+dq]=[pq]

ap+bq=p
and cp+dq=q
i.e., (a1)p+bq=0
and cp+(d1)q=0
Since [pq][00], the above system of equations have non-trivial solution.
So, a1bcd1=0

(a1)(d1)bc=0
ad(a+d)+1bc=0
adbc=a+d1
adbc=21=1

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