Question:
If $ A $ and $ B $ are two square matrices such that $ AB $ = $ A $ and $ BA $ = $ B $ , then
If $ A $ and $ B $ are two square matrices such that $ AB $ = $ A $ and $ BA $ = $ B $ , then
Updated On: Jun 14, 2022
- $ A $ and $ B $ are idempotent
- only $ A $ is idempotent
- only $ B $ is idempotent
- None of the above
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The Correct Option is A
Solution and Explanation
Since, $AB = A \,\,\,...(i)$
$\Rightarrow (AB) A = A \cdot A = A^2$
$\Rightarrow A(BA) = A^2$
[By associativity of matrix multiplication]
$\Rightarrow AB = A^2 \,\,[\because BA = B]$
$\Rightarrow A = A^2 \,\,$[From E$(i)$]
Since, $BA = B \,\,\,...(ii)$
$\Rightarrow (BA) B = B \cdot B = B^2$
$ B(AB) = B^2$
(By associativity of matrix multiplication]
$\Rightarrow BA = B^2 \,\,[\because AB = A]$
$\Rightarrow B = B^2 \,\,$ [From E $(ii)$]
Thus, $AA$ is equal to $A$ and $BB$ is equal to $B$.
Hence, $A$ and $B$ are idempotent.
$\Rightarrow (AB) A = A \cdot A = A^2$
$\Rightarrow A(BA) = A^2$
[By associativity of matrix multiplication]
$\Rightarrow AB = A^2 \,\,[\because BA = B]$
$\Rightarrow A = A^2 \,\,$[From E$(i)$]
Since, $BA = B \,\,\,...(ii)$
$\Rightarrow (BA) B = B \cdot B = B^2$
$ B(AB) = B^2$
(By associativity of matrix multiplication]
$\Rightarrow BA = B^2 \,\,[\because AB = A]$
$\Rightarrow B = B^2 \,\,$ [From E $(ii)$]
Thus, $AA$ is equal to $A$ and $BB$ is equal to $B$.
Hence, $A$ and $B$ are idempotent.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.