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Byju's Answer

Standard IX

Mathematics

Product Law

Let A and B b...

Question

Let A and B be two matrices such that AB $=$ A and BA $=$ B. Which of the following statements are correct?
$1$ . $A^{2} = A$
$2$ . $B^{2} = B$
$3$ . $(A B)^{2} = A B$
Select the correct answer using the code given below.

1

and

2

only

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2

and

3

only

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1

and

3

only

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1, 2

and

3

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Solution

The correct option is A $1$ and $2$ only
Given $A B = A B A = B$
${(A B)}^{2} = A^{2}$
$A^{2} = {(A B)}^{2}$
$= A (B A) B$
$= A (B) (B)$
$= (A B) B$ $= (A B) = A$
$A^{2} = A$ correct
$B^{2} = {(B A)}^{2}$
$= B (A B) A$
$= B A A$
$= (B A) A$
$= B A$
$= B$
$\Rightarrow B^{2} = B$ correct
${(A B)}^{2} = A^{2} = A$
${(A B)}^{2} \neq A B$
$∴$ Option A is true

Suggest Corrections

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Select the correct answer using the code given below :

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Select the correct answer using the code given below

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Let A and B be two matrices such that AB=A and BA=B. Which of the following statements are correct?1. A2=A2. B2=B3. (AB)2=ABSelect the correct answer using the code given below.

The correct option is A 1 and 2 onlyGiven AB=ABA=B(AB)2=A2A2=(AB)2=A(BA)B=A(B)(B)=(AB)B=(AB)=AA2=A correctB2=(BA)2=B(AB)A=BAA=(BA)A=BA=B⇒B2=Bcorrect(AB)2=A2=A(AB)2≠AB∴ Option A is true

Let A and B be two matrices such that AB $=$ A and BA $=$ B. Which of the following statements are correct?
$1$ . $A^{2} = A$
$2$ . $B^{2} = B$
$3$ . $(A B)^{2} = A B$
Select the correct answer using the code given below.