Question

If AB = A and BA = B , then which of the following is / are true?
$
  {\text{A}}{\text{. A is idempotent}} \\
  {\text{B}}{\text{. B is idempotent}} \\
  {\text{C}}{\text{. }}{{\text{A}}^T}{\text{ is idempotent}} \\
  {\text{D}}{\text{. none of these}} \\
$

Answer 1

Hint:- To solve this question first we have to understand what is an idempotent matrix. And then we have to start from given and proceed towards checking whether A or B is idempotent. That means on multiplying with itself either it yields the same or not.

Complete step-by-step solution -
First we have to understand idempotent matrix
Idempotent matrix:
In linear algebra an idempotent matrix is a matrix in which when multiplied by itself yields itself. That is the matrix is idempotent if and only if . for this product to be defined must necessarily be a square matrix.
Now we have given
AB = A and BA = B
Now we have to proceed from given
AB = A
Now on multiplying by matrix B on both side we get,
$B \times \left( {AB} \right) = B \times A$
 Now we can write it as
$\left( {BA} \right)B = BA$
Now as given in question BA = B , using this we get,
$
  B \times B = B \\
  \therefore {B^2} = B \\
 $
Hence B is an idempotent matrix because on multiplying by itself it yields itself.
Now we proceed from BA = B
On multiplying by matrix A on both side, we get
$A \times \left( {BA} \right) = A \times B$
We can write it as:
$\left( {AB} \right) \times A = AB$
As given in question AB = A using this we get,
$
  A \times A = A \\
  \therefore {A^2} = A \\
$
Hence A is an idempotent matrix because on multiplying with itself it yields itself.
Similarly
${\left( {{A^T}} \right)^2} = {A^T} \times {A^T} = {\left( {A \times A} \right)^T} = {\left( {{A^2}} \right)^T} = {\left( A \right)^T}$
Hence ${A^T}$ is also an idempotent matrix because on multiplying with itself it yields itself.
Hence option A , B , and C all are the correct options.

Note:- Whenever we get this type of question the key concept of solving is first we have to understand all the terms used in question. And we should have knowledge of algebra of matrices to solve this type of question. To solve this type of questions remember all the properties of matrices and determinants.