Perform matrix addition, subtraction, scalar multiplication and matrix multiplication, applying the order and conformability rules correctly.

What this dot point is asking

You must know the order of a matrix, perform the four operations correctly, and state when an operation is or is not defined.

Order and notation

A matrix is a rectangular array of numbers. Its order is rows $\times$ columns. The matrix

has order $2 \times 3$ (2 rows, 3 columns). The entry $a_{ij}$ sits in row $i$, column $j$, so $a_{23} = 4$.

Addition, subtraction and scalar multiplication

These three operations act on each element separately. Same-position entries are combined; for scalar multiplication every entry is multiplied by the scalar.

Matrix multiplication

To find the entry in row $i$, column $j$ of the product $AB$, multiply the elements of row $i$ of $A$ by the matching elements of column $j$ of $B$ and add the results.

If $A$ is $m \times n$ and $B$ is $n \times p$, the product $AB$ has order $m \times p$. The two inner numbers must match; the two outer numbers give the size of the answer.

Why order matters

Because multiplication combines rows of the first matrix with columns of the second, swapping the order changes both whether the product is defined and what it equals. In the costing example, $PQ$ is also defined ($2\times1$ times $1\times2$) but gives a $2 \times 2$ matrix, which means something entirely different.