Answer to Question #176376 in Linear Algebra for lyrad

Question

A and B are real non-zero 3 × 3 matrices and satisfy the equation (AB)^T + B^(-1) A = 0. Prove that if B is orthogonal then A is antisymmetric.

Answer 1

Using property of orthogonal matrix:

$$B^{-1}=B^T$$

we have

$$(AB)^T+B^{-1}A=B^TA^T+B^TA$$

$$=B^T(A^T+A)=0$$

Since $B$ is non-zero, product is zero when:

$$A^T+A=0$$

$$A=-A^T$$

Therefore in this case $A$ is antisymmetric.