[Math] Is it possible for a non-singular matrix to not have an inverse

Question

I know that a singular matrix cannot have an inverse, but I was curious as to whether or not a matrix with a non-zero determinant can have an inverse. And from what I understand the requirement for a matrix to be non-singular is that it has a determinant that is not zero.

Answer 1

Every square matrix $\mathbf{A}$ has an adjugate matrix $\textbf{adj}(\mathbf{A})$ formed from the cofactors of $\mathbf{A}$, even if $\mathbf{A}$ is singular. The inverse of a square matrix is the matrix

\begin{equation} \mathbf{A}^{-1}=\dfrac{\textbf{adj}\mathbf{(A)}}{\textbf{det}\mathbf{(A)}} \tag{1} \end{equation}

Since, if $\mathbf{A}$ is nonsingular, then its determinant is non-zero, every non-singular square matrix $\mathbf{A}$ has an inverse defined by equation $(1)$.

Wikipedia reference: https://en.wikipedia.org/wiki/Adjugate_matrix