Question

$A$ is a square matrix of order $n$.
$\ell =$ maximum number of distinct entries if $A$ is a triangular matrix
$m=$ maximum number of distinct entries if $A$ is a diagonal matrix
$p=$ minimum number of zeroes if $A$ is a triangular matrix
If $\ell +5=p+2m$, find the order of the matrix.

Answer 1

$\ell =1+2+3+......+n$
$\Rightarrow \ell =\frac{n(n+1)}{2}+1$ ;
$m=n+1,$ $p=\frac{n(n-1)}{2}$
Given, $l+5=p+2m\Rightarrow n=4$