UPSC NDA II 2024 Mathematics Paper

Concept:Matrix multiplication of a skew‑symmetric matrix $\rm P$ with a symmetric matrix $\rm Q$ yields a null matrix, and such products commute when both are zero.Explanation:Given $\rm P = \begin{bmatrix}0 & c & -b\\ -c & 0 & a\\ b & -a & 0\end{bmatrix}$ (skew‑symmetric) and $\rm Q = \begin{bmatrix}a^2 & ab & ac\\ ab & b^2 & bc\\ ac & bc & c^2\end{bmatrix}$ (symmetric).Compute $\rm PQ$ by multiplying row by column:Each entry $(\rm PQ)_{ij}$ equals the dot product of the $i$‑th row of $\rm P$ and the $j$‑th column of $\rm Q$.For example, $(1,1)$ entry: $0\cdot a^2 + c\cdot ab + (-b)\cdot ac = a(cb - bc)=0$.Similarly, all nine entries become zero. Hence $\rm PQ$ is a null matrix of order $3$.Compute $\rm QP$ likewise: each entry is the dot product of a row of $\rm Q$ and a column of $\rm P$, e.g., $(1,1)$: $a^2\cdot0 + ab\cdot(-c) + ac\cdot b = a(-bc+bc)=0$.Thus $\rm QP$ is also a null matrix, not an identity matrix. So statement II is false.Since $\rm PQ$ and $\rm QP$ are both null matrices, they are equal; statement III is true.Answer:Only statements I and III are correct. Hence Option C.