If A is a square matrix then A A^T is a) a diagonal matrix b) Symmetric matrix c) Skew symmetric matrix d) Scalar matrix

To determine the nature of the matrix $A-A^T$, let us analyze its properties.

Let $B=A-A^T$. Then, the transpose of $B$ is: $$B^T=(A-A^T{)}^T=A^T-(A^T{)}^T=A^T-A=-(A-A^T)=-B.$$ This shows that $B^T=-B$, which is the defining property of a skew-symmetric matrix.

Additionally, the diagonal elements of $B$ satisfy: $$B_{ii}=(A-A^T{)}_{ii}=A_{ii}-(A^T{)}_{ii}=A_{ii}-A_{ii}=0.$$ Thus, $B$ is a skew-symmetric matrix with zero diagonal elements.

Answer: c) Skew-symmetric matrix