JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

If $A$ and $B$ are symmetric matrices of the same order, then $A B - B A$ is a

Choose the correct answer:

Explanation

Step 1: Di gayi condition

Kyonki $A$ aur $B$ symmetric matrices hain, iska matlab:

A^T = A

B^T = B

Step 2: Let $X = AB - BA$

Iska nature check karne ke liye hum iska transpose ( $X^T$ ) nikalenge:

X^T = (AB - BA)^T

Step 3: Transpose ki properties apply karein

Hum jaante hain ki $(A - B)^T = A^T - B^T$ aur $(AB)^T = B^T A^T$ :

X^T = (AB)^T - (BA)^T

X^T = B^T A^T - A^T B^T

Step 4: Symmetric ki values rakhein

Upar di gayi $A^T = A$ aur $B^T = B$ ki values yahan rakhne par:

X^T = BA - AB

Step 5: Negative sign common lein

X^T = -(AB - BA)

X^T = -X

Conclusion:

Jab kisi matrix ka transpose uske negative ke barabar hota hai ( $X^T = -X$ ), toh use Skew-symmetric matrix kehte hain.

Final Answer:

$AB - BA$ ek Skew-symmetric matrix hai.