Algebraic Operations on Matrices

Three algebraic operations are defined on matrices.

(1) Scalar multiplication. For $A=[a_{ij}]_{m\times n}$ and a number (scalar) $k$, the matrix $kA=[k\,a_{ij}]_{m\times n}$ is obtained by multiplying every entry of $A$ by $k$. Taking $k=-1$ gives $-A=[-a_{ij}]$, the negative of $A$ — note this is not called "a negative matrix" (its entries need not all be negative).

(2) Addition and subtraction. If $A$ and $B$ have the same order, their sum $A+B=[a_{ij}+b_{ij}]$ is formed by adding corresponding entries, and it has that same order. Subtraction is defined via $A-B = A+(-1)B = [a_{ij}-b_{ij}]$. If $A$ and $B$ do not have the same order, $A+B$ and $A-B$ are simply not defined — there is no 'padding with zeros'. Addition/subtraction extends naturally to any finite number of same-order matrices.

(3) Multiplication. $A$ is conformable for the product $AB$ only when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, the product $AB$ exists and has order $m\times p$: $(m\times n)(n\times p)\to(m\times p)$ — the inner pair $n,n$ must match and "cancels", leaving the outer pair as the answer's order. The $(i,j)$ entry of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$:

For two row/column vectors, e.g. $[1\ 2\ 3]\begin{pmatrix}-2\\3\\5\end{pmatrix} = 1(-2)+2(3)+3(5)=[19]$, a $1\times1$ matrix (a single number). …