Question

If for square matrices A and B, AB=A and BA=B, prove A²=A and B²=B.

Answer 1

Answer:

If for square matrices A and B, AB=A and BA=B, prove A²=A and B²=B.

Given:

A and B are square matrices and

AB = A ........(1)

BA = B.........(2)

A²= A.A

A²= (AB).(AB)      (using (1))

A²= A(BA)B         (Matrix multiplication is associative)

A²= (AB)B            (using (2))

A²= AB                (using (1))

A²= A                  (using (1))

$\boxed{A^2=A}$

B²= B.B

B²= (BA).(BA)             (using (2))

B²= B(AB)A               (Matrix multiplication is associative)

B²= (BA)A                 (using (1))

B²= BA                     (using (2))

B²= B                       (using (2))

$\boxed{B^2=B}$

Hence proved