Matrix

When a linear system with many equations is written out, the unknowns and signs repeat endlessly; the only thing that changes from one system to another is the numbers. That is why nineteenth-century mathematicians decided to keep just those numbers in a rectangular table and to operate with the whole table. That table is the matrix, and over time it stopped being a mere shorthand and became an object with its own arithmetic.

The notation \(a_{ij}\) is an address: first the row, then the column. It reads much like a spreadsheet cell, and the order is worth memorizing because an \(m\times n\) matrix has \(m\) rows and \(n\) columns, and mixing them up is a frequent source of errors when multiplying. Two matrices are equal only if they have the same size and every entry agrees; a \(2\times3\) matrix is never equal to a \(3\times2\) one, even with the same numbers.

A guiding example: the system \(x+2y=5,\ 3x-y=1\) is summarized by the coefficient matrix \(\begin{pmatrix}1&2\\3&-1\end{pmatrix}\). Adding matrices entry by entry is natural, but the product, which comes later, is subtler. Think of a matrix as a machine that transforms vectors: that point of view justifies almost everything that follows. Two matrices can only be added when they have the same size, and they are equal when all their entries agree.