Rank 1 Matrix

Rank = Dimension

$$A=\left[\begin{array}{ccc}1& 4& 5\\ 2& 8& 10\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]\left[\begin{array}{ccc}1& 4& 5\end{array}\right].$$

A has Rank $1$ because each of its columns is a multiple of the first column.

Every rank 1 matrix $A$ can be written $A=\mathbf{u}{\mathbf{v}}^{\mathrm{T}}$, where $\mathbf{u}$ and $\mathbf{v}$ are column vectors. We can use rank $1$ matrices as building blocks for more complex matrices.