Choosing independent entries in a symmetric matrix So, the question is how many entries can be chosen indepently in a symmetric matrix of order n?
2) How many entries can be chosen indepently in a skew-symmetric matrix
$$
K^T=-K
$$
of order n. The diagonal of K is zero.
My attempt :
First of all I didn't understand what is meant by "how many entries can be chosen indepently"
And I also didn't understand "a matrix of order n". Is this the same as saying n by n Matrix?
Help me out on these concerns please. I believe I'm facing a linguistic problem in this case :)
 A: Due to the symmetry condition, choosing a value for some matrix element generally fixes the value of some other matrix element. Therefore that other matrix element cannot be chosen independently. For example, take a symmetric $2\times 2$ matrix,
$$M=\begin{pmatrix}*&*\\*&*\end{pmatrix}, M^T=M$$
And say you choose a value for the upper right element (i.e. $M_{12})$, say the value $2$:
$$M=\begin{pmatrix}*&2\\*&*\end{pmatrix}$$
Then the symmetry condition tells you that
$$M = \begin{pmatrix}*&2\\*&*\end{pmatrix} = M^T = \begin{pmatrix}*&*\\2&*\end{pmatrix}$$
ant thus
$$M = \begin{pmatrix}*&2\\2&*\end{pmatrix}$$
So you see that this choice also fixes the choice of element $M_{21}$, that is, $M_{21}$ cannot be chosen independently of $M_{12}$.
And yes, a "matrix of order $n$" is an $n\times n$ matrix.
A: By imposing $K^T = -K$, we have
$$K_{ij} = -K_{ji} \quad \forall i,j.$$
In particular, when $i = j$, 
$$K_{ii} = -K_{ii} \implies K_{ii} = 0.$$
By saying "entry can be chosen independently", we mean that an entry can be viewed an independent variable of others. For example, if $K_{12}$ is chosen to be an independent variable, then $K_{21}$ cannot be an independent variable since it depends on $K_{12}$, i.e.
$$K_{21} = -K_{12}.$$
Also note that the diagonal entries cannot be chosen independently since they must be zero. A natural choice of independent variables would be all entries in the strictly upper triangular part. Each entry in the strictly lower triangular part will be dependent on one entry in the strictly upper triangular part. Therefore the number of entries that can be chosen independently is
$$(n-1) + (n-2) + \ldots + 1 = \frac{1}{2}n(n-1).$$
An equaivalent formulation of the problem is the nullity of the linear transformation $T: \mathbb{M}_{n \times n} \to  \mathbb{M}_{n \times n}$ defined by
$$T(K) = K + K^T.$$
