Question

If AB = A and BA = B, where A and B are square matrices, then
$\left( a \right){B^2} = B{\text{ and }}{A^2} = A$
$\left( b \right){B^2} \ne B{\text{ and }}{A^2} = A$
$\left( c \right){B^2} = B{\text{ and }}{A^2} \ne A$
$\left( d \right){B^2} \ne B{\text{ and }}{A^2} \ne A$

Answer 1

Hint: In this particular question we use the concept that if A, and B are square matrices then from the associative property of matrix, A (BA) = (AB) A, so use this concept to reach the solution of the question.

Complete step-by-step answer:
Given data:
AB = A..................... (1)
And, BA = B................... (2), where A, and B are the square matrices.
Form equation (1) we have,
AB = A
Now from equation (2) substitute the value of B in the above equation we have,
Therefore, A (BA) = A
Now from associative property of matrix, if A and B are square matrix then, A (BA) = (AB) A, so use this property in the above equation we have,
Therefore, A (BA) = A
Therefore, (AB) A= A (associative property of matrix)
Now substitute the value of AB from equation (1) in the above equation we have,
$ \Rightarrow {A^2} = A$
Now form equation (2) we have,
BA = B
Now from equation (1) substitute the value of A in the above equation we have,
Therefore, B (AB) = B
Now from associative property of matrix, if A and B are square matrix then, B (AB) = (BA) B, so use this property in the above equation we have,
Therefore, (BA) B = B (associative property of matrix)
Now substitute the value of BA from equation (2) in the above equation we have,
$ \Rightarrow {B^2} = B$
So this is the required answer.
Hence option (a) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that we always recall the basic properties of matrices such as associative property, distributive property etc. Associative property is the key to the above problem, so use this property as above applied and simplify we will get the required result.