Question

If $AB = A$ and $BA = B$, then ${B^2}$ is equal to
1. $B$
2. $A$
3. $ - B$
4. ${B^3}$

Answer 1

Hint: In this question, we are given that $AB = A$ and $BA = B$. We have to find the value of ${B^2}$. The first step is to let the given terms A and B be the square matrices. Now, we will post multiply $B$ in the second equality condition. Then, again and again, use the given equality conditions and you’ll get the value.

Formula Used:
Let, the given equation of matrix be $PQ = P$
Pre- multiplication by $P$ will be; $P \times PQ = P \times P$
Similarly post- multiplication of $P$ is $PQ \times P = P \times P$

Complete step by step Solution:
Let, A and B are the square matrices,
Given that,
$AB = A - - - - - \left( 1 \right)$
And $BA = B - - - - - \left( 2 \right)$
Now, taking equation (2) and post multiplying both sides by $B$
$BA \times B = B \times B$
$B\left( {AB} \right) = {B^2}$
From equation (1)
$BA = {B^2}$
From equation (2)
$B = {B^2}$
It can be written as ${B^2} = B$

Hence, the correct option is 1.

Note: The key concept involved in solving this problem is the good knowledge of pre- and post-multiplication. Students must remember that if here we were asked the value of A. Then, the first step will be the pre-multiplication of A in equation (1). And again, repeating the given conditions continuously until we will get the value. Pre multiplication and post multiplication is the concept of Matrix. In the matrix we can’t change the series as we do in normal so, we follow this. Also, the first matrix must have the same number of columns just as the second matrix has rows in order to perform matrix multiplication. The number of rows in the resulting matrix equals the number of rows in the original matrix, and the number of columns equals the number of columns in the original matrix.