# Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as:

A matrix is just a compact notation, which allows you to specify several linear equations at once without having to write them all out. For example, instead of writing

  3x + 4y + 5z =  7

2x - 3y + 6z =  6

2x + 5y - 9z = 11


I can write the same thing more compactly using matrices:

$$\begin{bmatrix} 3 & 4 & 5\\ 2 & -3 & 6\\ 2 & 5 & -9\\ \end{bmatrix} \begin{bmatrix}x\\ y\\ z\\ \end{bmatrix} = \begin{bmatrix}7\\ 6\\ 11\\ \end{bmatrix}$$

However, it seems that the more obvious way to do it would be:

$$\begin{array}{ll} \begin{bmatrix} x & y & -z\\ \end{bmatrix} \\ \begin{bmatrix} 3 & 4 & 5\\ 2 & -3 & 6\\ 2 & 5 & -9\\ \end{bmatrix} = \begin{bmatrix}7\\ 6\\ 11\\ \end{bmatrix} \end{array}$$

Why are matrices written in a seemingly more complex way than the alternative?

What's the advantage of doing so?

• I think they are written like in the first example because when you multiply two matrices you make the scalar product of a row of the first matrix with a column of the second. In your case you have your matrix $\begin{pmatrix}3&4&5\\2&-3&6\\2&5&-9\end{pmatrix}$ multiplied by another matrix ($3\times 1$) $\begin{pmatrix}x\\y\\z\end{pmatrix}$. Matrix multiplication is,usually, written as in your first example. – Bman72 Sep 9 '14 at 6:33
• Writing a linear system is not the only use of matrices. What's the common sense way for a product of three matrices? – Jean-Claude Arbaut Sep 9 '14 at 6:33

The real answer is that Dr. Math (and your teacher) aren't telling you everything there is to know about matrices. There's a lot more to linear algebra than expressing systems of linear equations. A matrix is a handy way to represent a "linear transformation," which is a function with some special properties. Solving equations like the one you posted is essentially the same thing as finding a vector $v$ so that $T(v)=w$, where $T$ is the linear transformation, and $w$ is the thing you have on the right side of the equation.
Notice that in the notation above, $T$ looks like a function being applied to $v$. That's because it is. If we think of $T$ as a matrix, and $v$ as the vector $\left(\substack{x\\ y\\ z}\right)$, then we want to mirror the function notation, so we write the vector to the right of the matrix.