Question

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

Answer 1

A singular matrix $A$ is a square matrix which is not invertible, i.e. $A^{-1}$ does not exist. A non-singular matrix $B$ is a square matrix which is invertible, i.e. $B^{-1}$ exists.

Further, since for a non-singular matrix $B$, we have $BB^{-1} = B^{-1} B = I$ and since, for $B^{-1} B$ to exist, the number of columns in $B$ must be equal to the number of rows in $B^{-1}$ , and also, for $B^{-1} B$ to exist, the number of columns in $B^{-1}$ must be equal to the number of rows in $B$ , hence the number of columns in $B$ must be equal to the to the number of rows in $B$ i.e. $B$ must be a square matrix.