Answer to Question #112840 in Linear Algebra for kgodiso

Question

what is the difference between a singular and a non singular matrix,pls show that a non singular matrix must be square

Answer 1

Singular matrix :

A n×n square matrix A is a singular matrix if its determinant is zero. In other words, a n×n square matrix is a singular matrix if it does not have an inverse A⁻¹. For example :

Let us take a 2×2 square matrix A as $\begin{bmatrix} 12 & 8 \\ 3 & 2 \end{bmatrix}$

Here , determinant of matrix A = 12x2-3x8=0

So , inverse of matrix A i.e. A^-1  does not exist.

Hence, A is a singular matrix.

Non- singular matrix :

A n×n square matrix P is a non-singular matrix if its determinant is non-zero. In other words, a n×n square matrix is a non-singular matrix if its inverse P⁻¹ exists. For example :

Let us take a 2×2 square matrix P as $\begin{bmatrix} 10 & 3 \\ 5 & 2 \end{bmatrix}$

Here , determinant of matrix P = 10x2-3x5=5 which is not equal to zero.

So , inverse of matrix P i.e. P^-1  exists.

Hence, P is a non-singular matrix.

Now , to prove that non-singular matrix must be square:

In the definition itself, a non-singular matrix is a square matrix .

Because if suppose that a matrix A is not a square matrix , then, it is not possible to find the determinant of this matrix A and determine whether the determinant is zero or non-zero to decide its singularity.

Hence , is non- singular matrix is a square matrix .