If B is a skew-symmetric matrix, write whether the matrix AB A T is symmetric or skew-symmetric.

Question

Philoid · Accepted Answer

B is a skew – symmetric matrix, then

B^T = -B …(i)

Consider

(ABA^T)^T = (A^T)^T B^T A^T

[∵ (AB)^T = B^TA^T]

⇒ (ABA^T)^T = AB^TA^T [∵ (A^T)^T = A]

⇒ (ABA^T)^T = A(- B)A^T [from (i)]

⇒ (ABA^T)^T = - ABA^T

∴ ABA^T is a skew – symmetric matrix