UPSC NDA I 2025 Mathematics Paper

Rotation matrix at θ=π :
f(π)=[

−1	0
0	−1

]
To find (f(π))2, multiply the matrix by itself:
f(π)×f(π)=[

−1	0
0	−1

]×[

−1	0
0	−1

]
Using matrix multiplication:
Top-left element: (−1)(−1)+(0)(0)=1
Top-right element: (−1)(0)+(0)(−1)=0
Bottom-left element: (0)(−1)+(−1)(0)=0
Bottom-right element: (0)(0)+(−1)(−1)=1
Resulting matrix:
f(π)2=[

1	0
0	1

]
∴(f(π))2 is equal to the identity matrix, which is [

1	0
0	1

].