If \( \mathrm{AB}=\mathrm{A} \) and \( \mathrm{BA}=\mathrm{B} \), where \( \mathrm{A} \) and \( \mathrm{B} \) are square matrices of same order, then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \mathrm{B}^2=\mathrm{B},\ \mathrm{A}^2=\mathrm{A} \)
Since AB = A and BA = B, a factor can be replaced by the given product, and the brackets moved by associativity of matrix multiplication.
\( \displaystyle \mathrm{A}^2=\mathrm{A}\mathrm{A} \)
\( \displaystyle =(\mathrm{AB})\mathrm{A} \)
\( \displaystyle =\mathrm{A}(\mathrm{BA}) \)
\( \displaystyle =\mathrm{A}\mathrm{B}=\mathrm{A} \)
\( \displaystyle \mathrm{B}^2=\mathrm{B}\mathrm{B} \)
\( \displaystyle =(\mathrm{BA})\mathrm{B} \)
\( \displaystyle =\mathrm{B}(\mathrm{AB}) \)
\( \displaystyle =\mathrm{B}\mathrm{A}=\mathrm{B} \)
So \( \mathrm{A}^2=\mathrm{A} \) and \( \mathrm{B}^2=\mathrm{B} \); both matrices are idempotent.