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Let $A$ and $B$ be two matrices such that $A B=A$ and $B A=B$. Which of the following statements are correct? [2014-II]
$1.$ $A^{2}=A$
$2.$ $\quad B^{2}=B$
$3.$ $\quad(A B)^{2}=A B$
Select the correct answer using the code given below:
Options:
$1.$ $A^{2}=A$
$2.$ $\quad B^{2}=B$
$3.$ $\quad(A B)^{2}=A B$
Select the correct answer using the code given below:
Solution:
1171 Upvotes
Verified Answer
The correct answer is:
1,2 and 3
We have, $\mathrm{AB}=\mathrm{A}$
$\therefore \quad \mathrm{A}^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A} .(\mathrm{BA}) \mathrm{B}$
=ABB $\quad(\because \mathrm{BA}=\mathrm{B})$
$=\mathrm{AB}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$
Also, $B^{2}=(B A) \cdot(B A)=B(A B) \cdot A$
$=$ B.A.A $\quad(\because \mathrm{AB}=\mathrm{A})$
$=$ B.A $=B$ $\quad(\because \mathrm{BA}=\mathrm{B})$
Again, $(\mathrm{AB})^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A}$. (BA) B
$=$ A.B.B $\quad(\because \mathrm{BA}=\mathrm{B})$
$=$ A. $\mathrm{B}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$
$\therefore \quad \mathrm{A}^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A} .(\mathrm{BA}) \mathrm{B}$
=ABB $\quad(\because \mathrm{BA}=\mathrm{B})$
$=\mathrm{AB}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$
Also, $B^{2}=(B A) \cdot(B A)=B(A B) \cdot A$
$=$ B.A.A $\quad(\because \mathrm{AB}=\mathrm{A})$
$=$ B.A $=B$ $\quad(\because \mathrm{BA}=\mathrm{B})$
Again, $(\mathrm{AB})^{2}=(\mathrm{AB}) \cdot(\mathrm{AB})=\mathrm{A}$. (BA) B
$=$ A.B.B $\quad(\because \mathrm{BA}=\mathrm{B})$
$=$ A. $\mathrm{B}=\mathrm{A}$ $\quad(\because \mathrm{AB}=\mathrm{A})$
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