Exercise 1.3.3

Question

Prove that $(a A+b B)^{t}=a A^{t}+b B^{t}$ for any $A, B \in \mathrm{M}_{m 	imes n}(F)$ and any $a, b \in F$.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
Let $M=aA+bB$ and $N=aA^t+bB^t$. Then we have $$M_{ij}=aA_{ij}+bB_{ij}=N_{ji}$$
and so $M^t=N$.
\end{proof}

Answers