> [!Definition] Definition (Inverse of Real Square Matrix) > Let $A$ be a [[Real Square Matrices|square matrix with real entries]] of order $n.$ We say that $B$ is an of $A$ if the [[Matrix Product|product]] $BA=I_{n}=AB$, where $I_{n}$ is the [[Real Identity Matrix|identity matrix]] of order $n$: that is, $B$ is both a [[Left Inverse of Real Matrix|left-]] and [[Right Inverse of Real Matrix|right-inverse]] of $A.$ **Language**: If a square matrix $A$ has an inverse then we say that $A$ is **invertible**. Otherwise, we say that $A$ is **singular**. # Properties > [!NOTE] Corollary (Zero row or column implies not invertible) > Suppose $E\in \mathbb{F}^{m\times m}$ is invertible then $E$ does not have a zero row or column. > >Proof. WLOG if row $1$ is a zero row then have $(1,0,\dots,0) E=(0,0,\dots,0)$. Post multiplying by $E^{-1}$ gives $(1,0,\dots,0)=(0,0,\dots,0)E^{-1} = \underline{0}$which leads to a contradiction.