If A and B are two matrices such that AB=B and BA=a, then find the value of A 2 +B 2
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(A+B).(A+B)=A^2+AB+BA+B^2
=A^2+B^2+AB+BA
because AB=B and BA=A
A^2+B^2=(A+B). (A+B)-A-B
or
A^2=A.A=A (BA)=(AB)A=BA=A
in the same way
B^2=B.B=B(AB)=(BA).A=BA=B
hence
A^2+B^2=A+B
=A^2+B^2+AB+BA
because AB=B and BA=A
A^2+B^2=(A+B). (A+B)-A-B
or
A^2=A.A=A (BA)=(AB)A=BA=A
in the same way
B^2=B.B=B(AB)=(BA).A=BA=B
hence
A^2+B^2=A+B
abhi178:
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