Matrix Operations

By the end of this lesson, you will be able to add, subtract, and multiply matrices; identify matrix dimensions; apply scalar multiplication; and understand when matrix multiplication is defined and why it is not commutative.

The Big Idea

A matrix is a rectangular array of numbers — a powerful structure for organizing data and solving systems. Matrix operations are the language of linear algebra, used in economics, computer graphics, machine learning, physics, and statistics. The non-commutativity of matrix multiplication ($AB \neq BA$ in general) is a critical conceptual shift from number arithmetic.

Matrix Notation

An $m \times n$ matrix has $m$ rows and $n$ columns.

$$A = \begin{pmatrix} 2 & -1 & 3 \ 0 & 5 & -2 \end{pmatrix} \quad \text{is a } 2 \times 3 \text{ matrix}$$

Element $a_{ij}$ is in row $i$, column $j$.

Addition and Subtraction

Add (or subtract) corresponding elements. Matrices must have the same dimensions.

$$\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & -1 \ 0 & 3 \end{pmatrix} = \begin{pmatrix} 6 & 1 \ 3 & 7 \end{pmatrix}$$

Scalar Multiplication

Multiply every element by the scalar.

$$3 \begin{pmatrix} 2 & -1 \ 4 & 0 \end{pmatrix} = \begin{pmatrix} 6 & -3 \ 12 & 0 \end{pmatrix}$$

Matrix Multiplication

$A \cdot B$ is defined only when columns of $A$ = rows of $B$.

If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.

Element $(i, j)$ of $AB$: dot product of row $i$ of $A$ and column $j$ of $B$.

$$c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$$

Example 1 — Matrix Multiplication

$$\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}$$

$(1,1)$: $1(5) + 2(7) = 19$

$(1,2)$: $1(6) + 2(8) = 22$

$(2,1)$: $3(5) + 4(7) = 43$

$(2,2)$: $3(6) + 4(8) = 50$

Result: $\begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$

Non-Commutativity

$$AB \neq BA \text{ in general}$$

Example: $\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \ 4 & 3 \end{pmatrix}$

but

$\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \ 1 & 2 \end{pmatrix}$

Different results. Order matters.

Identity Matrix

The $n \times n$ identity matrix $I_n$ has 1s on the diagonal and 0s elsewhere:

$$I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$