Linear systems formulas

Applies to any $m\times n$ matrix $A$ after row reduction; the rank is the same in every echelon form.

Equivalent statement: $\mathbf{b}$ lies in the column space of $A$. Always $\mathrm{rank}(A)\le \mathrm{rank}([A\mid \mathbf{b}])\le \mathrm{rank}(A)+1$.

When the system is consistent, ${\mathbf{x}}_p$ is any one particular solution; the general solution adds an arbitrary null-space vector.

$n$ is the number of columns of $A$ (not rows). Nullity equals the number of free variables in an echelon form of $A$.

For $A$ of size $m\times n$ and any product $AB$ that is defined. For real matrices also $\mathrm{rank}(A^{T}A)=\mathrm{rank}(A{A}^T)=\mathrm{rank}(A)$.

First (Sylvester's inequality): $A$ is $m\times n$ and $B$ is $n\times p$. Second: $A$ and $B$ have the same size. Third: $P$ ($m\times m$) and $Q$ ($n\times n$) are invertible. Also, for non-zero $\mathbf{u}\in {\mathbb{R}}^m$ and $\mathbf{v}\in {\mathbb{R}}^n$, the $m\times n$ matrix $\mathbf{u}{\mathbf{v}}^T$ has rank 1.

For a square $n\times n$ matrix, with $I$ the $n\times n$ identity. If the left block cannot be reduced to $I$ (a zero row appears in it), $A$ is singular and has no inverse.

For a square $n\times n$ system with $\det A\ne 0$. $A_i$ is $A$ with its $i$-th column replaced by $\mathbf{b}$.