MCQ
If $AB = A$ and $BA = B$, where $A$ and $B$ are square matrices, then :
- ✓$B^2=B$ and $A^2=A$
- B$B^2 \neq B$ and $A^2=A$
- C$A^2 \neq A, B^2=B$
- D$A^2 \neq A, B^2 \neq B$
✓
Answer
Correct option: A.
$B^2=B$ and $A^2=A$
Here,
$A B=A \cdot .(1)$
$B A=B \ldots(2)$
$\Rightarrow A B A=A A\ [$ Multiplying both sides by $A]$
$B A B=B B\ [$ Multiplying both sides by $ A]$
$\Rightarrow A B=A^2\ [$ From eq. $(2)]$
$B A=B^2\ [$ From eq. $(1)]$
$\Rightarrow A=A^2\ [$ From eq. $(1)]$
$B=B^2\ [$From eq. $(2)]$
$A B=A \cdot .(1)$
$B A=B \ldots(2)$
$\Rightarrow A B A=A A\ [$ Multiplying both sides by $A]$
$B A B=B B\ [$ Multiplying both sides by $ A]$
$\Rightarrow A B=A^2\ [$ From eq. $(2)]$
$B A=B^2\ [$ From eq. $(1)]$
$\Rightarrow A=A^2\ [$ From eq. $(1)]$
$B=B^2\ [$From eq. $(2)]$
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