A square matrix of order n is non-singular if its determinant is non zero and therefore its rank is n. Its all rows and columns are linearly independent and it is invertible.
The concept of nonsingular matrix is for square matrix, it means that the determinant is nonzero, and this is equivalent that the matrix has full-rank.
Nonsingular means the matrix is in full rank and you the inverse of this matrix exists.
Dear Gunasekaran Nallusamy,
A square matrix $A$ of order $n$ is said to be nonsingular if and only if there is a matrix $B =: A^{-1}$ such that $AA^{-1} = A^{-1}A = I_{n}$. The matrix
$A^{-1}$ is said to be the inverse of $A$.
From the inverse formula
$A^{-1} = (1/det A)adj A$.
We see that, if $det A = 0$ then $A$ has no inverse (singular matrix). And by definition of rank of matrix:
Definition. A non-zero matrix $A$ is said to have rank $r$ if at least one of its $r$-square minors is different from zero while every $(r + 1)$-square minor, if any, is zero. A zero matrix is said to have rank $0$.
Therefore A is nonsingular if and only if $rank A = n$. Equivalently, $A$ has no inverse if and only if $rank A$ less than $n$.
Non-singular Matrix - square matrix (n by n), full rank matrix (dimension - n), invertible and determinant is non-zero.
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