# symmetric matrices multiplications

I have two square matrices $P$ and $Q$. Given $P = I - PQ$ I managed to show that $P$ is invertible and that $PQ = QP$. I need to show somehow that if $Q$ is symmetric then $P$ is symmetric as well.

I have tried to develop $(I - PQ)^T$ in hopes that it will equal to $P$ but no such luck.

$$P(I+Q)=I$$ Taking transpose: $$(I+Q)P^T=I$$
Are you able to see why $P$ is symmetric?
• I see. what is the definition of $(I+Q)$ being invertible? – Siong Thye Goh Nov 17 '17 at 9:50
• a more proper term is to multiply $(I+Q)^{-1}$. We can post multiply something for one equation and pre multiply something for another equation. – Siong Thye Goh Nov 17 '17 at 10:14
• @eventhorizon02 Alternatively, we can use the two equalities given in this answer to say $$P = PI = P(I+Q)P^T = IP^T = P^T$$ – Arthur Nov 17 '17 at 11:04