Q.Assertion (A) : Matrix $A = \begin{bmatrix}0 & -6 & 7\\6 & 5 & -1\\-7 & 1 & 0\end{bmatrix}$ is a skew-symmetric matrix. Reason (R) : A matrix A is skew-symmetric if $A' = -A$. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is $not$ the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

For a square matrix $A$, the transpose $A'$ (also written $A^{T}$) swaps rows and columns, so the $(i,j)$ entry of $A'$ is the $(j,i)$ entry of $A$.

• $A$ is symmetric if $A' = A$; equivalently $a_{ij} = a_{ji}$, so the matrix is a mirror image across its main diagonal.
• $A$ is skew-symmetric if $A' = -A$; then $a_{ij} = -a_{ji}$, which forces every diagonal entry to be $0$.

A key result used repeatedly in problems: every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix, …