Search any question & find its solution
Question:
Answered & Verified by Expert
If $A$ and $B$ are two matrices such that $A B=A$ and $B A=B$, then which one of the following is correct?
Options:
Solution:
1757 Upvotes
Verified Answer
The correct answer is:
$\left(A^{T}\right)^{2}=A^{T}$
Let $A$ and $B$ be two matrices such that $A B=A$ and $B A$ $=B$
Now, consider $A B=A$ Take Transpose on both side $(A B)^{T}=A^{T}$
$\Rightarrow A^{T}=B^{T} . A^{T}$ ...(1)
Now, $\mathrm{BA}=\mathrm{B}$
Take, Transpose on both side $(B A)^{T}=\mathrm{B}^{\mathrm{T}}$
$\Rightarrow \mathrm{B}^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}} \cdot \mathrm{B}^{\mathrm{T}}$ ...(2)
Now, from equation (1) and (2). we have $A^{T}=\left(A^{T} \cdot B^{T}\right) A^{T}$
$A^{T}=A^{T}\left(B^{T} A^{T}\right)$
$=A^{T}(A B)^{T} \quad\left(\because(A B)^{T}=B^{T}=B^{T} A^{T}\right)$
$=A^{T} \cdot A^{T}$
Thus, $A^{T}=\left(A^{T}\right)^{2}$
Now, consider $A B=A$ Take Transpose on both side $(A B)^{T}=A^{T}$
$\Rightarrow A^{T}=B^{T} . A^{T}$ ...(1)
Now, $\mathrm{BA}=\mathrm{B}$
Take, Transpose on both side $(B A)^{T}=\mathrm{B}^{\mathrm{T}}$
$\Rightarrow \mathrm{B}^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}} \cdot \mathrm{B}^{\mathrm{T}}$ ...(2)
Now, from equation (1) and (2). we have $A^{T}=\left(A^{T} \cdot B^{T}\right) A^{T}$
$A^{T}=A^{T}\left(B^{T} A^{T}\right)$
$=A^{T}(A B)^{T} \quad\left(\because(A B)^{T}=B^{T}=B^{T} A^{T}\right)$
$=A^{T} \cdot A^{T}$
Thus, $A^{T}=\left(A^{T}\right)^{2}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.