To create a special matrix !!

Question

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$.
Where $N$ is a scalar and $\mathbb{I}_N$ is the $N \times N$ identity matrix.
Examle for 2x2 matrix
$-1$ $1$
$1$ $1$

Example for 4x4 matrix
$-1$ $1$ $1$ $1$
$1$ $-1$ $1$ $1$
$1$ $1$ $-1$ $1$
$1$ $1$ $1$ $-1$

Answer 1

For some $N$, this is known to be impossible. For some other specific values of $N$, there are known ways of construction. Yet, in general, no one knows how to do it or if such a matrix exists.

See Hadamard matrix on Wikipedia.