Algebra of Matrices — Addition, Subtraction, Scalar Multiplication and Transpose

Matrices of the same order can be combined by ordinary arithmetic, entry by entry.

Addition and Subtraction

If $A=(a_{ij})$ and $B=(b_{ij})$ are matrices of the same order $m\times n$, then

Addition/subtraction is not defined for matrices of different orders — this is a common source of error, so always check the order first.

Matrix addition is commutative ($A+B=B+A$) and associative ($(A+B)+C=A+(B+C)$), exactly like ordinary number addition, because it reduces to ordinary addition of the corresponding real-number entries.

Scalar Multiplication

Multiplying a matrix $A=(a_{ij})$ by a real number (scalar) $k$ multiplies every element by $k$:

This is used, for example, to scale an entire sales or cost matrix by a uniform percentage increase.

Transpose of a Matrix

The transpose of a matrix $A$, written $A'$ (or $A^{T}$), is obtained by interchanging its rows and columns: the $i$-th row of $A$ becomes the $i$-th column of $A'$. If $A$ has order $m\times n$, then $A'$ has order $n\times m$. Two useful facts: $(A')'=A$, and $(A+B)'=A'+B'$.

Symmetric and Skew-Symmetric Matrices

A square matrix $A$ is:

• symmetric if $A'=A$ (elements are mirror-symmetric about the leading diagonal, $a_{ij}=a_{ji}$);
• skew-symmetric if $A'=-A$ (which forces every diagonal element to be $0$, since $a_{ii}=-a_{ii}$ only when $a_{ii}=0$).

Every square matrix $A$ can be written as the sum of a symmetric part and a skew-symmetric part:

$A = \tfrac{1}{2}(A+A') + \tfrac{1}{2}(A-A')$ …