Question

Consider the following statements:
1. If $A$ and $B$ are two square matrices of same order, then $(A+B)(A-B)=A^2-B^2$.
2. If $A$ and $B$ are two square matrices of same order, then $(AB{)}^n=A^n{B}^n$.
3. If $A$ and $B$ are two matrices such that $AB=A$ and $BA=B$, then $A$ and $B$ are idempotent.
Which of these is/are not correct?

• A. $\left(1\right)and\left(2\right)$
• B. $\left(2\right)and\left(3\right)$
• C. $\left(3\right)and\left(1\right)$
• D. All of these

Answer 1

Answer: (A)

$\left(1\right)and\left(2\right)$

1. Given : $A$ and $B$ are square matrices of same order

To check : $(A+B)(A-B)=A^2-B^2$

$=A(A-B)+B(A-B)$

$=A^2-AB+BA-B^2$

$\ne A^2-B^2$ ($\because$ Matrix multiplication is not commutative $\therefore AB\ne BA$)

2. Given : $A$ and $B$ are square matrices of same order

To check : $(AB{)}^n=A^n{B}^n$

Consider : $(AB{)}^n$

$=\left(AB\right)\left(AB\right)......\left(AB\right)$

ntimes

$=A\left(BA\right)\left(BA\right)......\left(BA\right)B$

$\ne A^n{B}^n$ ($\therefore$ Matrix Multiplication is not commutative $\therefore BA\ne AB$)

3. Given : $AB=A,BA=B$

To check: $A$ and $B$ are idemponent

i.e To check $A^2=A,B^2=B$

$=AB=A$ (Given) ......(1)

multiply both sides by $A$

$\left(AB\right)A=AA$

$A\left(BA\right)=A^2$ (Associativity)

$AB=A^2$.......(2)

Similarly, by Post multiplying $B$ on both sides of $BA=B$,

we get $B^2=B$

$\therefore \left(1\right)and\left(2\right)arenotcorrect$