If AB = A and BA = B, then which of the following is/are true?

# If AB = A and BA = B, then which of the following is/are true?

1. A

A is idempotent

2. B

B is idempotent

3. C

AT is idempotent

4. D

none of these

Fill Out the Form for Expert Academic Guidance!l

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

### Solution:

Given, AB = A, BA = B

or (BA)B = B or B2 = B

Also,

$\begin{array}{r}\mathrm{A}×\mathrm{BA}=\mathrm{AB}\\ ⇒\left(\mathrm{AB}\right)\mathrm{A}=\mathrm{A}\\ ⇒{\mathrm{A}}^{2}=\mathrm{A}\end{array}$

Now ${\left({\mathrm{A}}^{\mathrm{T}}\right)}^{2}=\left({\mathrm{A}}^{\mathrm{T}}×{\mathrm{A}}^{\mathrm{T}}\right)=\left(\mathrm{A}×\mathrm{A}{\right)}^{\mathrm{T}}={\left({\mathrm{A}}^{2}\right)}^{\mathrm{T}}={\mathrm{A}}^{\mathrm{T}}$

Similarly, (BT)2 = BT

Hence, AT and BT are idempotent.

## Related content

 Area of Square Area of Isosceles Triangle Pythagoras Theorem Triangle Formulae Perimeter of Triangle Formula Area Formulae Volume of Cone Formula Matrices and Determinants_mathematics Critical Points Solved Examples Type of relations_mathematics

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)