Let $m$ be the order of a Hadamard matrix. The matrix $[1]$ shows that order 1 is possible, and the parent 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 multiplying selected columns by $-1$ Then permute columns as needed to arrive at a matrix whose first three rows have the following form, where $P$ denotes a submatrix of one row and all 1's and $N$ denotes a submatrix of one row and all $-1$ s.

$$\begin{matrix} \begin{matrix} x \quad &\quad y & \quad z & \quad w \end{matrix} & \begin{matrix} \quad \end{matrix} \\ \left[ \begin{matrix} \overbrace{P} & \overbrace{P} & \overbrace{P} & \overbrace{P} \\ P & P & N & N \\ P & N & P & N \\ \end{matrix} \right] \end{matrix} $$

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