Matrix Addition & Scalar Multiplication

Overview

Matrix addition and scalar multiplication are the fundamental operations that allow us to combine and scale matrices. These operations define the vector space structure on $M_{m \times n}$.

Matrix Addition

For matrices $A$ and $B$ of the same size ($m \times n$):

$A + B = [a_{ij} + b_{ij}]$

Example

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Requirement

Matrices must have the same dimensions to be added.

Properties of Matrix Addition

Property	Formula
Commutative	$A + B = B + A$
Associative	$(A + B) + C = A + (B + C)$
Identity	$A + O = A$
Inverse	$A + (-A) = O$

Scalar Multiplication

For scalar $c$ and matrix $A$:

$cA = [c \cdot a_{ij}]$

Example

$3 \times \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times 2 \\ 3 \times 3 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix}$

Properties of Scalar Multiplication

Property	Formula
Distributive (scalar)	$c(A + B) = cA + cB$
Distributive (matrix)	$(c + d)A = cA + dA$
Associative	$c(dA) = (cd)A$
Identity	$1A = A$
Zero	$0A = O$

Matrix Subtraction

Defined as addition of the negative:

$A - B = A + (-1)B = A + (-B)$

Example

$\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 5-1 & 6-2 \\ 7-3 & 8-4 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}$

Linear Combinations of Matrices

A linear combination of matrices $A_1, A_2, \ldots, A_k$:

$c_1 A_1 + c_2 A_2 + \cdots + c_k A_k$

Example

$2\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + 3\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 2 \end{bmatrix}$

The Zero Matrix

The $m \times n$ zero matrix $O_{m \times n}$ has all entries equal to 0.

Properties:

• $A + O = O + A = A$
• $A - A = O$
• $0A = O$
• $OA = O$ (for matrix multiplication, when defined)

Negative of a Matrix

The negative of $A$ is $(-1)A$:

$-A = [-a_{ij}]$

Example

$-\begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ -3 & 4 \end{bmatrix}$

Worked Examples

Example 1: Combined Operations

Given:

$A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}, \quad c = 2$

Find $2A - 3B$:

$2A = \begin{bmatrix} 4 & -2 \\ 6 & 0 \end{bmatrix}, \quad 3B = \begin{bmatrix} 3 & 9 \\ 6 & 3 \end{bmatrix}$

$2A - 3B = \begin{bmatrix} 4-3 & -2-9 \\ 6-6 & 0-3 \end{bmatrix} = \begin{bmatrix} 1 & -11 \\ 0 & -3 \end{bmatrix}$

Example 2: Solve for Matrix

Find $X$ if $2X + A = B$ where:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Solution:

$2X = B - A = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}$

$X = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$