Matrix Operations: Addition and Scalar Multiplication
The foundational entry-by-entry operations that turn matrices into a vector-space structure.
1. Matrix Addition
Definition (Entry-wise Addition)
Let $A = [a_{ij}]$ and $B = [b_{ij}]$ both be $m \times n$ matrices. Their sum is defined entry by entry:
$$ A + B = [a_{ij} + b_{ij}] $$That is, the $(i,j)$-entry of $A+B$ is simply $a_{ij} + b_{ij}$.
⚠️ Compatibility Requirement (Law): Addition is only defined when $A$ and $B$ have identical dimensions. If $A$ is $m \times n$ and $B$ is $p \times q$ with $m \neq p$ or $n \neq q$, then $A + B$ is undefined. No exceptions — you cannot add a $2 \times 3$ matrix to a $2 \times 2$ matrix.
Concrete Example
Let
$$ A = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 \\ -1 & 2 \end{bmatrix} $$Both are $2 \times 2$, so addition is defined:
$$ A + B = \begin{bmatrix} 1+4 & -2+5 \\ 3+(-1) & 0+2 \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 2 & 2 \end{bmatrix} $$Incompatible Pair — Rejected
Let
$$ C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2 \times 3}, \quad D = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix}_{2 \times 2} $$$C$ is $2 \times 3$ and $D$ is $2 \times 2$. The column counts differ ($3 \neq 2$), so $C + D$ is undefined.
Subtraction via the Opposite Matrix
The additive inverse (or opposite) of $A$ is $-A = (-1)A = [-a_{ij}]$, obtained by negating every entry. Then:
$$ A - B = A + (-B) $$For the matrices above:
$$ A - B = \begin{bmatrix} 1-4 & -2-5 \\ 3-(-1) & 0-2 \end{bmatrix} = \begin{bmatrix} -3 & -7 \\ 4 & -2 \end{bmatrix} $$Notice the entry $(2,1)$: $3 - (-1) = 3 + 1 = 4$. Subtracting a negative entry adds — a common source of sign errors.