A square matrix A = [aᵢⱼ]ₙₓₙ is called a diagonal matrix if aᵢⱼ = 0 for

The correct option is i ≠ j.

To understand why this is the correct option, let us review the definition of a diagonal matrix.
A square matrix $A=[a_{ij}{]}_{n\times n}$ is a matrix in which the number of rows is equal to the number of columns (both equal to $n$).

The elements $a_{ij}$ for which $i=j$ (such as $a_{11}$, $a_{22}$, $a_{33}$, ..., $a_{nn}$) constitute the principal diagonal (or main diagonal) of the matrix.

By definition, a square matrix is called a diagonal matrix if all its non-diagonal elements are equal to zero.
The non-diagonal elements are those elements where the row index $i$ is not equal to the column index $j$.

Therefore, the mathematical condition for $A$ to be a diagonal matrix is:
$a_{ij}=0\text{for all}i\ne j$
This matches the correct option i ≠ j.