Question

Why does multiplying \[A+A^T\] by \[\frac{1}{2}\] not change its symmetric property?

Options

• Because multiplication by \[\frac{1}{2}\] makes every matrix skew-symmetric.

• Because \[(kA)^T=kA^T\].

• Because \[(kA)^T=-kA^T\].

• Because \[(A+A^T)^T=-(A+A^T)\].

Answer 1

For a scalar \[k\], the transpose rule is \[(kA)^T=kA^T\]. Therefore a scalar multiple of a symmetric matrix remains symmetric, including \[\frac{1}{2}(A+A^T)\].