COMEDK 2025 Morning Solved Paper

Concept:The product of $A(t)$ and $A(-t)$ is the identity matrix because $A(t)$ and $A(-t)$ represent rotation by $t$ and $-t$, which cancel each other.Explanation:First, compute $A(-t)$. Since $\cos(-t) = \cos t$ and $\sin(-t) = -\sin t$, we have:$A(-t) = \begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{bmatrix}$.Now multiply $A(t)$ by $A(-t)$:$A(t)A(-t) = \begin{bmatrix} \cos t & \sin t \\ -\sin t & \cos t \end{bmatrix} \begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{bmatrix}$.Calculate each entry:• First row, first column: $\cos t \cdot \cos t + \sin t \cdot \sin t = \cos^2 t + \sin^2 t = 1$.• First row, second column: $\cos t \cdot (-\sin t) + \sin t \cdot \cos t = -\cos t \sin t + \sin t \cos t = 0$.• Second row, first column: $(-\sin t) \cdot \cos t + \cos t \cdot \sin t = -\sin t \cos t + \cos t \sin t = 0$.• Second row, second column: $(-\sin t) \cdot (-\sin t) + \cos t \cdot \cos t = \sin^2 t + \cos^2 t = 1$.Thus the product is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, the identity matrix.Answer:Option A: Identity matrix.