Reading matrices and combining entries

Rows, columns and order

A matrix is a rectangular array. Its order is the number of rows followed by the number of columns: a \(2\times3\) matrix has two rows and three columns. An entry is identified by its row first, then its column. A row matrix has one row; a column matrix has one column. A square matrix has equal numbers of rows and columns.

Equality and entrywise operations

Equal matrices have the same order and equal corresponding entries. Add or subtract matrices only when their orders match. Multiply a matrix by a scalar by multiplying every entry, including any negative or zero entry.

For example, \(2\begin{pmatrix}1&-3\\ 0&4\end{pmatrix}=\begin{pmatrix}2&-6\\ 0&8\end{pmatrix}\). The zero matrix has zero in every position. A square identity matrix has ones on its main diagonal and zeros elsewhere; it acts as an identity under a compatible matrix product.

Keep row and column meanings in the same order when matrices represent data. Adding a row of Monday quantities to a row of Tuesday quantities is meaningful only if their columns refer to the same items and units.