Matrices & Determinants
Rotation Matrix — Truth of Statements
nta_pyq_2024_jan
Grade 12
Question:
Consider the matrix $f(x)=\begin{bmatrix}\cos x&-\sin x&0\\\sin x&\cos x&0\\0&0&1\end{bmatrix}$. Given below are two statements: Statement I: $f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below
Statement I is false but Statement II is true
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Both Statement I and Statement II are true
Step-by-Step Solution
Key Concept: Statement I: $f(-x)$ has $\cos x$ on diagonal and $\sin x,-\sin x$ off diagonal — this is $f(x)^{-1}=f(x)^T$. Indeed $f(x)\cdot f(-x)=I$. ✓ Statement II: Multiply $f(x)\cdot f(y)$ using trig addition formulas: result is $f(x+y)$. ✓
$f(x)\cdot f(-x)=I$ (rotation matrix property). $f(x)\cdot f(y)=f(x+y)$ by angle addition. Both statements are true.
Correct Answer: 4