Let A,B,C be 33 matrices such that A is symmetric and B,C are skew-symmetric.

Question

Let $A,B,C$ be $3	imes3$ matrices such that $A$ is symmetric and $B,C$ are skew-symmetric. Consider the statements:

(S1) $A^{13}B^{26}-B^{26}A^{13}$ is symmetric.

(S2) $A^{26}C^{13}-C^{13}A^{26}$ is symmetric.

Accepted Answer

$A^{T}=A, B^{T}=-B, C^{T}=-C$

$$
\begin{aligned}
P & =A^{13} B^{26}-B^{26} A^{13} \\
P^{T} & =\left(A^{13} B^{26}-B^{26} A^{13}\right)^{T}=\left(A^{13} B^{26}\right)^{T}-\left(B^{26} A^{B}\right)^{T} \\
& =\left(B^{26}\right)^{T}\left(A^{13}\right)^{T}-\left(A^{13}\right)^{T}\left(B^{26}\right)^{T} \\
& =\left(B^{T}\right)^{26}\left(A^{T}\right)^{13}-\left(A^{T}\right)^{13}\left(A^{T}\right)^{26} \\
& =B^{26} A^{13}-A^{13} B^{26}=-\left(A^{13} B^{26}-B^{26} A^{13}\right)=-P
\end{aligned}
$$

$P$ is skew-symmetric matrix $\Rightarrow S_{1}$ is false

$Q=A^{26} C^{13}-C^{13} A^{26}=Q^{T}=\left(A^{26} C^{13}-C^{13} A^{26}\right)^{T}$

$Q=\left(A^{26} C^{13}\right)^{T}-\left(C^{13} A^{26}\right)^{T}=\left(C^{13}\right)^{T}\left(A^{26}\right)^{T}-\left(A^{26}\right)^{T}\left(C^{13}\right)^{T}$

$=\left(C^{T}\right)^{13}\left(A^{T}\right)^{26}-\left(A^{T}\right)^{26}\left(C^{T}\right)^{13}=-C^{13} A^{26}+A^{26} C^{13}$

$=A^{26} C^{13}+C^{13} A^{26}$

$\Rightarrow Q^{T}=Q \Rightarrow Q$ is symmetric matrix $\Rightarrow S_{2}$ is true.

Solutions · 1