# [Math] Is not full rank matrix invertible

determinantlinear algebralinear-transformationsmatricesmatrix-rank

## Problem

$$A$$ is a $$4 \times 4$$ matrix. It is known that $$\text{rank}(A)=3$$. Is matrix A invertible ?

## Attempt to solve

$$\text{rank(A)}=3 \implies \det(A)=0$$
which implies matrix is $$\textbf{not}$$ invertible. One dimension is lost during linear transformation if matrix is not full rank by definition. This implies determinant will be $$0$$ and that some information is lost in this linear transformation.

Is my intuition behind this correct ?

The following properties are equivalent for a square matrix $$A$$:
• $$A$$ has full rank
• $$A$$ is invertible
• the determinant of $$A$$ is non-zero