If AB = A and BA = B, where A and B are square matrices, then: - Vidyadip

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 $A^2 = A$

$B^2 = B$ and $A^2 = A$
Here,
$AB = A ...(1)$
$BA = B ...(2)$
$\Rightarrow \text{ABA} = AA \ [$Multiplying both sides by $A]$
$\text{BAB} = BB \ [$Multiplying both sides by $A]$
$\Rightarrow AB = A^2 \ [$From eq. $(2)]$
$BA = B^2 \ [$From eq. $(1)]$
$\Rightarrow A = A^2 \ [$From eq. $(1)]$
$B = B^2 \ [$From eq.$ (2)]$

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