proof that Hadamard matrix has order 1 or 2 or 4n

Let $m$ be the order of a Hadamard matrix. The matrix $[1]$ shows that order 1is possible, and the entry has a $2\times 2$ Hadamard matrix, so assume $m>2$.

We can assume that the first row of the matrix is all 1’s by multiplyingselected columns by $-1$. Then permute columns as needed to arrive at amatrix whose first three rows have the following form, where $P$ denotes a submatrix of one rowand all 1’s and $N$ denotes a submatrix of one row and all $-1$’s.

Since the rows are orthogonal and there are $m$ columns we have

Adding the 4 equations together we get

$$4x=m.$$

so that $m$ must be divisible by 4.