Addition and Scalar Multiplication of Matrices

Addition of Matrices

Definition. If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of the same order $m\times n$, their sum $A+B$ is the matrix obtained by adding corresponding entries:

Addition is defined only when the two matrices have identical order; $A+B$ is simply not defined if the orders differ, since there would be no way to pair up entries.

Scalar Multiplication

Definition. If $A=[a_{ij}]$ is a matrix of order $m\times n$ and $k$ is any real number (a scalar), the scalar multiple $kA$ is the matrix obtained by multiplying every entry of $A$ by $k$:

The negative of a matrix, $-A$, is the special case $k=-1$; subtraction $A-B$ is then defined as $A+(-B)$, i.e. entrywise subtraction of corresponding entries. Both operations are used together in Example 4 and Exercise: Matrix Addition and Scalar Multiplication, Q2, where an expression such as $3A-2B$ is evaluated by first scaling each matrix separately and then adding entrywise.

Properties of Addition and Scalar Multiplication

For matrices $A,B,C$ of the same order $m\times n$ and scalars $k,l$, the following hold (each can be checked by writing out the general entry $a_{ij}$ of both sides and comparing, since ordinary addition and multiplication of real numbers already satisfy these laws):

• Commutativity of addition: $A+B=B+A$ (Exercise: Matrix Addition and Scalar Multiplication, Q4 verifies this numerically).
• Associativity of addition: $(A+B)+C=A+(B+C)$, so a sum of three or more matrices may be bracketed in any order.
• Additive identity: $A+O=O+A=A$, where $O$ is the zero matrix of the same order as $A$.
• Additive inverse: $A+(-A)=O$; every matrix has a unique additive inverse, namely the matrix of its negated entries.
• Distributivity over matrix addition: $k(A+B)=kA+kB$.
• Distributivity over scalar addition: $(k+l)A=kA+lA$. …