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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.

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It should not be that hard...no worries. –  1015 Feb 2 '13 at 16:01
    
I guess everybody here will be able to answer, but I somewhat feel that answering your question will deprive you of the chance to learn for youself. Why don't you tell us which definition you are using and what you have to check exactly for every single question. I bet you will find some answers yourself. Also: make sure you understand how you can consider a scalar as a matrix. What is $A'$ ? do you mean the transpose? –  Simon Markett Feb 2 '13 at 16:06

2 Answers 2

up vote 1 down vote accepted

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.

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I Finally reviewed my notes and understand what you're talking about...thank you so much. –  bryansis2010 Feb 13 '13 at 15:29
    
@bryansis2010 Great! You're welcome. –  1015 Feb 13 '13 at 15:31

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 $\,A'\,$...

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+1. Don. I have learned much from your answers here. –  Babak S. Feb 4 '13 at 9:19

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