Question

Which expression writes any square matrix \[A\] as the sum of a symmetric and a skew-symmetric matrix?

Options

• \[A=\frac{1}{2}(A+A^T)-\frac{1}{2}(A-A^T)\]

• \[A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)\]

• \[A=\frac{1}{2}(A-A^T)+\frac{1}{2}(A^T-A)\]

• \[A=(A+A^T)+(A-A^T)\]

Answer 1

The symmetric part is \[\frac{1}{2}(A+A^T)\], and the skew-symmetric part is \[\frac{1}{2}(A-A^T)\]. Adding them cancels the \[A^T\] terms and leaves \[A\].