Question

if ab=a and ba=b show that a and b are idempotent​

Answer 1

Idempotency means that an element gives the same result as itself however you impose functions and operations on it.

a and b satisfy the above condition.

Therefore, a and b are idempotent.

Answer 2

$\huge\pink{\boxed{\blue{\boxed{ \purple{ \boxed{{\pink{Answer}}}}}}}} \\ \large\pink{\boxed{\blue{\boxed{ \purple{ \boxed{{\pink{Your~answer↓}}}}}}}} \\ \small\bold\red{definition}$

Idempotent Matrix :- A square matrix A is called idempotent matrix if

$\small\bold\red{ {A}^{2} = A }$

Given:-

A and B are square matrices such that AB = A and BA = B.

To Prove :-

A and B are idempotent matrices.

Proof :-

Ist Part

As BA = B

Premultiply by A on both sides, we get

A(BA) = AB

(AB)A = A .......(as AB = A)

A.A = A

$\small\bold\red{ = > \: {A}^{2} = A }$

Hence, proved.

Proof of 2nd part

AB = A

Premultiply by B on both sides, we get

B(AB) = BA

(BA)B = B ..........(as BA = B)

B.B = B

$\small\bold\red{ = > \: {B}^{2} = B}$

Hence, proved.