Adding matrices and multiplying by a scalar

Addition and subtraction combine corresponding entries, so the matrices must have the same order. In a data model their row meanings, column meanings and units must also match. Multiplication by a scalar multiplies every entry.

Example

Let \(A=\begin{pmatrix}2&-1\\0&3\end{pmatrix}\) and \(B=\begin{pmatrix}1&4\\-2&5\end{pmatrix}\).

\(A+B=\begin{pmatrix}3&3\\-2&8\end{pmatrix}\), and \(A-B=\begin{pmatrix}1&-5\\2&-2\end{pmatrix}\).

\(\begin{aligned}2A-B&=\begin{pmatrix}4&-2\\0&6\end{pmatrix}-\begin{pmatrix}1&4\\-2&5\end{pmatrix}\\&=\begin{pmatrix}3&-6\\2&1\end{pmatrix}.\end{aligned}\)

Subtract negative entries carefully: \(0-(-2)=2\). A \(2\times3\) matrix cannot be added to a \(3\times2\) matrix even though each has six entries.