# Is there anything special with a 3x3 matrix where the 3rd row is 0 0 1?

I'm coding using p5.js and I'm looking at this method https://p5js.org/reference/#/p5/applyMatrix

Using that method, I can multiply my current matrix with any matrix of the form:

$$\begin{pmatrix} a & c & e \\ b & d & f \\ 0 & 0 & 1 \\ \end{pmatrix}$$

by calling applyMatrix(a, b, c, d, e, f)

There is no method for multiplying any arbitrary matrix like: $$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix}$$

Is there anything special with a matrix of that form? Is it possible to convert any arbitrary matrix (like the bottom matrix) into a matrix of that form?

• Do you have a question about math? – John Douma Jan 18 '19 at 3:34
• My question is about matrices not the coding itself, I just put the link there for context. – DarkPotatoKing Jan 18 '19 at 3:39
• Your question appears to be about some programming language. – John Douma Jan 18 '19 at 3:39
• You could fit your $3 \times 3$ matrix into the larger matrix $$\pmatrix{1&2&3&0\\4&5&6&0\\7&8&9&0\\0&0&0&1}$$ which I would say is a "matrix of that form" – Ben Grossmann Jan 18 '19 at 3:42
• @DarkPotatoKing It is used to represent affine transformations. (This is also hinted at in the page you linked.) – Alex Provost Jan 18 '19 at 3:45

It is a standard way to represent an affine transformation of the plane; this is how it is used on the page you linked. The submatrix $$A = \begin{pmatrix} a & c \\ b & d \end{pmatrix}$$ in your question represents the linear part of the affine transformation, and the extra column $$t = \begin{pmatrix} e \\ f \end{pmatrix}$$ to the right corresponds to the translation part of the transformation. In full, the corresponding transformation maps a vector $$v$$ to the vector $$Av + t$$.