Matrix transpose, how do I see this?

Question

I am unable to see this identity.

Let a in $R^n$, $B$ in $R^{n,n}$, c in $R^n$

$c^TBa + a^TBc = a^T(B^T+B)c$.

How do I see this?

Answer 1

Since $a^TBc$ is a $1\times1$ matrix, it's equal to its own transpose. But$$(a^TBc)^T=c^TB^Ta.$$Can you take it from here?

Answer 2

$(c^tBa)^t = a^tB^tc$ and so $c^tBa + a^tBc = a^tB^tc+a^tBc = a^t(B^t+B)c$, since the matrix product $c^tBa$ is only a value and so taking the transpose doesn't matter.