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,
$AB = A ...(1)$
$BA = B ...(2)$
$\Rightarrow ABA = AA \ [$Multiplying both sides by $A]$
$BAB = BB \ [$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 $eq. (2)]$
$AB = A ...(1)$
$BA = B ...(2)$
$\Rightarrow ABA = AA \ [$Multiplying both sides by $A]$
$BAB = BB \ [$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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