Let abcdef F be scalars and suppose that A and B are the fol

Let a,b,c,d,e,f F be scalars, and suppose that A and B are the following matrices: A = [a b 0 c] and B B = [d e 0 f] Prove that AB = BA if and only if det ([b a - c e d - f]) = 0.

finding AB value

AB = [ ad ae+bf]

[0 cf]

finding BA

BA = [ ad db+ec]

[0 fc]

So for AB=BA

ae+bf must be equals to db+ec

ae+bf = db +ec ------>1

find the det condtion value

b(d-f) - (a-c)e =0

bd -bf- ea +ce =0

bd +ce = bf+ea ---------------->2

so 1 and 2 are equal

so hence proved

Let a,b,c,d,e,f F be scalars, and suppose that A and B are the following matrices: A = [a b 0 c] and B B = [d e 0 f] Prove that AB = BA if and only if det ([b

Let abcdef F be scalars and suppose that A and B are the fol

Solution

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