Equality, Addition, Subtraction and Scalar Multiplication of Matrices

Equality of matrices

Two matrices are equal only if (a) they are of the SAME order, and (b) every corresponding element is equal. For example, $\begin{pmatrix}2&5\\1&3\end{pmatrix}=\begin{pmatrix}x&5\\1&y\end{pmatrix}$ forces $x=2$ and $y=3$ — matrix equality is really shorthand for several ordinary equations, one per matching pair of elements.

Addition and subtraction — conformability

Two matrices can be added or subtracted only if they are of the SAME order — called being conformable for addition/subtraction. Adding a $2\times2$ matrix to a $2\times3$ matrix is undefined.

When $A$ and $B$ are of the same order, $A+B$ (or $A-B$) is found by adding (or subtracting) each pair of CORRESPONDING elements:

Matrix addition is commutative ($A+B=B+A$) and associative ($(A+B)+C=A+(B+C)$), because each element position is simply an independent ordinary addition.

Scalar multiplication

When a matrix $A$ is multiplied by an ordinary number $k$ (a scalar), every SINGLE element of $A$ is multiplied by $k$: …