Matrix Algebra - True or False? I have 5 T-F statements right here about symmetric matrices, and to the right are my attempts. I have a feeling some of them are wrong, though.
(a) Symmetric matrices must be square. (T)
There can be a rectangular matrix which is symmetrical.
(b) A scalar is symmetric. (T)
I don't know why
(c) If $A$ is symmetric, then $ \alpha A$  is symmetric. (T)
What is $\alpha A$?
(d) The sum of symmetric matrices is symmetric.
I think this is false as not all sums will be symmetric.
(e) If $(A')' = A$ , then  $A$ is symmetric.  
I think this is true.
Could I encourage that, if you know one part of the answer, please go ahead to answer them? I know that it will be very, very hard to find one answer with all 5.
 A: Hints:
(a) This is true apparently directly from the very definition of symmetric matrix, but also  because one the main characteristics of a symmetric matrix $\,A\,$ is that $\,A^t=A\,$ , and this equality forces $\,A\,$ .
(b) You seem to have meant "a scalar matrix is symmetric", and a scalar matrix is of the form
$$\alpha I=\begin{pmatrix}\alpha&0&0&...&0\\0&\alpha&0&...&0\\...&...&...&...&...\\0&0&0&...&\alpha\end{pmatrix}$$
So what say you? Is that symmetric or not?
(c) $\,A=(a_{ij})\,$ is symmetric iff $\,a_{ij}=a_{ji}\,\,\forall\,i\neq j\,$ , and $\,\alpha A=(\alpha a_{ij})\,$ ...
(d) This now follows at once from the above, and about (e) I'm not sure what you mean by $\,A'\,$...
A: I'll take $A=A^t$ (where $A^t$ denotes the transposed matrix) for the definition of $A$ to be symmetric.
(a) True. If $A$ is an $m\times n$ matrix and is equal to its transpose which is an $n\times m$ matrix, then $m=n$.
(b) True. A scalar is a $1\times 1$ matrix, so equal to its transpose. More generally, what is the transpose of a scalar matrix? Itself.
(c) True. Note that $\alpha A$ is the matrix obtained from $A$ by multiplying each coefficient by $\alpha$.
(d) True (symmetric matrices of same size). Essentially because $(A+B)^t=A^t+B^t$.
(e) False. It $A'$ denotes $A^t$ the transpose of $A$.... Note that $(A^t)^t=A$ for every matrix $A$. And not every matrix is symmetric.
