Recall that a unit $M_{n \times n}(\mathbb{R})$ is a matrix that has a multiplicative inverse. Let $U$ be the set of all units of $M_{n \times n}(\mathbb{R})$. Prove $U$ is closed under multiplication. Prove $U$ is not closed under addition.
What I have in my proof is:
We will now prove $U$ is closed under multiplication. We will prove this directly. That is, we will show the multiplicative closure of $U$. We will use a $2 \times 2$ matrix, and some real numbers $a,b,c,d,e,f,g,h$ to get
\begin{array}{cc} a & b \\ c & d \\ \end{array}
and \begin{array}{cc} e & f \\ g & h \\ \end{array} We will multiply them to get \begin{array}{cc} a & b \\ c & d \\ \end{array}
\begin{array}{cc} e & f \\ g & h \\ \end{array}
\begin{array}{cc} ae+bg & af+bf \\ ce+dg & cf+dh \\ \end{array} Since $a,b,c,d,e,f,g,h$ are all real numbers and we know $ae+bg,af+bf,ce+dg,cf+dh$ are real number because real numbers are closed under addition and multiplication which proves $U$ has multiplicative closure, thus proving this part of the proof.
My prof has told me I have to prove this has a multiplicative inverse, and I must do this for all n x n matrices and not just 2 x 2 matrices. I am confused on how to do that so I was wondering if anyone could offer me assistance?
As for the addition part, I know I must add two matrices for $U$ and show the result does not have a multiplicative inverse, but I also do not know how to go about this. Can anyone please help?