Question

If Aand Bare symmetric n×nmatrices, then A B B Amust be symmetric as well.

Short answer

True

Step-by-step solution

Step by step solution Step 1: Properties of the transpose

Consider the properties of the transpose below.

1. $(A+B{)}^T=A^T+B^T$For all m× nmatrices Aand B.
2. $(kA{)}^T=k{A}^T$For all m×nmatrices Aand for all scalars k.
3. $(AB{)}^T=B^T{A}^T$ For all m×pmatrices Aand for all matrices B.
4. $\text{rank}\left(A^T\right)=\text{rank}\left(A\right)$For all matrices A.
5. $(A^T{)}^{-1}=(A^{-1}{)}^T$ For all invertible n×nmatrices A

Determine whether the statement is true or false

Consider the terms below.

$\begin{array}{rcl}(ABBA{)}^T& =& (BA{)}^T(AB{)}^T\\ & =& A^T{B}^T{B}^T{A}^T\\ & =& ABBA\end{array}$

Hence, the statement is true.