Answer to Question #115610 in Linear Algebra for Thabang

Question

Explain the difference between a singular and a non-singular matrix. Show that a non-singular
matrix must be square.

Answer 1

A nxn square matrix P is non-singular or invertible matrix if there exists a matrix B such that:

PB=BP=I

that is P has an inverse or in other words, determinant of P is non-zero.

A nxn square matrix Q is singular or non-invertible matrix if, determinant of Q is zero, that is Q does not have an inverse.

to show non-singular matrix be square:

let us assume that a non-singular matrix A is not a square matrix.

then, it is not possible to compute its inverse as it is not possible to find a determinant of a non-square matrix.

so, by definition, a non-singular matrix has to be a square matrix.