# If A and B are symmetric matrices, which of the following matrices is symmetric? a) A^2-B^2 b) (A+B)(A-B) c) AB*AB

If A and B are symmetric matrices, which of the following matrices is symmetric?
a) ${A}^{2}-{B}^{2}$
b) $\left(A+B\right)\left(A-B\right)$
c) $AB\cdot AB$
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Caylee Davenport
Given that A and B are symmetric matrices.
That is ${A}^{T}=A,{B}^{T}=B$
Consider,
$\left[\left(A+B\right)\left(A-B\right){\right]}^{T}=\left(A-B{\right)}^{T}\left(A+B{\right)}^{T}\phantom{\rule{0ex}{0ex}}=\left({A}^{T}-{B}^{T}\right)\left({A}^{T}+{B}^{T}\right)\phantom{\rule{0ex}{0ex}}=\left(A-B\right)\left(A+B\right)$
Not symmetric
Consider
$ABAB\phantom{\rule{0ex}{0ex}}\left[ABAB{\right]}^{T}={A}^{T}{B}^{T}{A}^{T}{B}^{T}\phantom{\rule{0ex}{0ex}}=ABAB$
which is symmetric.