JAMIA 2024 — Mathematics PYQ
JAMIA | Mathematics | 2024If A is a square matrix such that A2=A, then (I−A)3+A is equal to
Choose the correct answer:
- A.
1
(Correct Answer) - B.
0
- C.
I−A
- D.
I+A
1
Explanation
Solution:
Hume (I−A)3+A ki value nikalni hai, jahan A2=A diya gaya hai.
Step 1: (I−A)3 ka expansion karein
Binomial expansion (x−y)3=x3−3x2y+3xy2−y3 ka use karte hue:
Kyonki I (Identity matrix) ki koi bhi power I hi rehti hai aur IA=A hota hai:
Step 2: A2=A ka use karke simplify karein
Sawal mein diya hai ki A2=A. Iska matlab hai:
-
A2=A
-
A3=A2⋅A=A⋅A=A2=A
Ab expansion mein A2 aur A3 ki jagah A rakhne par:
Step 3: Final Expression ki value nikaalein
Ab hume sawal ke mutabiq (I−A)3+A ki value nikalni hai:
Explanation
Solution:
Hume (I−A)3+A ki value nikalni hai, jahan A2=A diya gaya hai.
Step 1: (I−A)3 ka expansion karein
Binomial expansion (x−y)3=x3−3x2y+3xy2−y3 ka use karte hue:
Kyonki I (Identity matrix) ki koi bhi power I hi rehti hai aur IA=A hota hai:
Step 2: A2=A ka use karke simplify karein
Sawal mein diya hai ki A2=A. Iska matlab hai:
-
A2=A
-
A3=A2⋅A=A⋅A=A2=A
Ab expansion mein A2 aur A3 ki jagah A rakhne par:
Step 3: Final Expression ki value nikaalein
Ab hume sawal ke mutabiq (I−A)3+A ki value nikalni hai: