# Matrices | meaning

Algebra was years ago and I cannot remember what kind of operation the | means. For example:

Where M is a 3x3 matrix and C a column-vector.

• I would guess is the concatenation of $M$ with (minus)the resulting vector $MC$. So $P$ is $3\times 4$. Sep 21, 2017 at 12:26
• But what is the concatenation? As I said, my matrices knowledge is long gone :X Sep 21, 2017 at 12:35
• @GonacFaria concatenation means adjoining them one after the other. For example concatenating the strings "far" and "out" would give me "farout" or "outfar" depending on the order of concatenation. Sep 21, 2017 at 13:03

Say we have $$M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} , \hspace{1cm} C = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ So $$-MC = \begin{pmatrix} -ax - by - cz \\ -dx - ey - fz \\ -gx - hy - iz \end{pmatrix}$$ Then $$P = [M | -MC] = \begin{pmatrix} a & b & c & (-ax - by - cz)\\ d & e & f & (-dx - ey - fz) \\ g & h & i & (-gx - hy - iz) \end{pmatrix} \in \mathbb{R}^{3 \times 4}$$
It is the notation for an augmented matrix. Namely, it is a 3 by 4 matrix where the last column is the column vector $\mathbf{-MC}$.