Question

Let A and B be 3 $\times$ 3 matrices of real numbers, where A is symmetric, B is skew-symmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(AB{)}^T=(-1{)}^{k}AB,$ where (AB)${}^t$ is the transpose of the matrix AB, then find the possible values of k.
Which values of k are acceptable?

• A. 5
• B. 6
• C. 7
• D. 8

Answer 1

Answer: (A), (C)

The correct options are
A 5
C 7

Given $A$ is symmetric, $B$ is skew-symmetric of order $3\times 3$ and $(A+B)(A-B)=(A-B)(A+B)$.
$\Rightarrow A^2-AB+BA-B^2=A^2+AB-BA-B^2$
$\Rightarrow AB=BA$
$(AB{)}^T=(-1{)}^{k}AB$
$\Rightarrow B^T{A}^T=(-1{)}^{k}AB$
$\Rightarrow -BA=(-1{)}^{k}AB\Rightarrow -AB=(-1{)}^{k}AB$
$\therefore (-1{)}^k=-1$
$\therefore k$ is an odd number.
Hence, possible options are A and C.