Question

If A and B are two matrices such that AB=B and BA=a, then find the value of A2+B2

Answer 1

☆☆heya your answer is here☆☆


Your mistake is that you have assumed that A and B are invertible, which does not have to be the case. 
A² + B² = A(BA) + B(AB) 
= (AB)A + (BA)B 
= BA + AB 
= A + B. 

It's certainly not true that A or B has to be the identity. For instance, they could both be the 0 matrix, or the matrix 
[1 0] 
[0 0]. 

The interesting question is whether there is a solution with B not equal to A. If we can show that B must always equal A, then your other solutions would be valid (though they can be simplified to 2A and 2B). 

However, this turns out not to be the case. Take A = 
[0 0] 
[a 1] 
and B = 
[0 0] 
[b 1] 
for any two different numbers a and b. Then AB = B and BA = A, but A² + B² is 
[0 0] 
[a+b 1] 
which is equal to A+B, but not equal to 2AB (=2B), or 2BA (=2A).

Answer 2

Answer:

A^2+B^2

=AA+BB

=A(BA)+B(AB)

=(AB)A+(BA)B

=BA+AB

=A+B

Hope this will help you

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