Question

If $AB = A$ and $BA = B$ , then:
A) ${A^2} = A$
B) ${B^2} = B$
C) $A = I$
D) $B = I$

Answer 1

Hint: In this question, we will make two equations which are already given in the question. Then we will put the value of B&A in equation $1 \& 2$ respectively. Thus we can see that A and B are idempotent matrix. Thus we will get the answer.

Complete step by step solution: Given that
$AB = A........\left( 1 \right)$
And
$BA = B.........\left( 2 \right)$

One thing we need to keep in mind while solving the matrics problems is we can’t cancel out them like numbers. it’s only possible when matrics are invertible. which is not given in our problem. so we’ll only use the given equations.
Put the value of $B$ from equation $2$ in equation $1$
$
  A\left( {BA} \right) = A \\
  AB\left( A \right) = A \\
 $
Now use the value of A from equation (1).
i.e.
$A.A = A$
Similarly,
Put the value of $A$ from equation $2$ in equation $1$
$
  BA = B \\
  B\left( {AB} \right) = B \\
 $
i.e.
$B.B = B$
That means A and B are idempotent matrices. From here
$A = I$
And $B = I$

Hence options C and D both are correct.

Note: First we have to remember what an idempotent matrices and how we denote it. A matrix is A is idempotent if ${A^2} = A$. After that, by putting the values given in the question within each other we get the correct answer, and it proves that $A$ and B are idempotent matrices and denoted by $I$.