Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are not just a bookkeeping tool for systems of equations — they are objects in their own right, with their own arithmetic. Matrix operations are the foundation of computer graphics, machine learning, engineering simulations, and economic modeling.

In this lesson, you will learn the rules for adding, subtracting, scaling, and multiplying matrices, including the critical fact that matrix multiplication is not commutative — the order you multiply matters.

Matrix Dimensions

A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix (read ”$m$ by $n$”). The dimensions are always stated rows first, columns second.

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$

This is a $2 \times 3$ matrix — 2 rows, 3 columns.

Individual entries are written $a_{ij}$, where $i$ is the row number and $j$ is the column number. In the matrix above, $a_{12} = 2$ (row 1, column 2) and $a_{21} = 4$ (row 2, column 1).

Matrix Addition and Subtraction

Two matrices can be added or subtracted only if they have the same dimensions. The operation is performed entry by entry.

If $A$ and $B$ are both $m \times n$ matrices, then:

$(A + B)_{ij} = a_{ij} + b_{ij}$

Example 1: Matrix Addition

$\begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix} + \begin{bmatrix} 4 & 0 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix} 1+4 & 3+0 \\ 2+(-2) & -1+5 \end{bmatrix} = \begin{bmatrix} 5 & 3 \\ 0 & 4 \end{bmatrix}$

Example 2: Matrix Subtraction

$\begin{bmatrix} 7 & 2 \\ 4 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 5 \\ 1 & 6 \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 3 & 3 \end{bmatrix}$

Undefined operations: You cannot add a $2 \times 3$ matrix to a $3 \times 2$ matrix. The dimensions must match exactly.

Scalar Multiplication

Multiplying a matrix by a scalar (a single number) means multiplying every entry by that number.

$k \cdot A = \begin{bmatrix} k \cdot a_{11} & k \cdot a_{12} \\ k \cdot a_{21} & k \cdot a_{22} \end{bmatrix}$

Example 3: Scalar Multiplication

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

Example 4: Linear Combination of Matrices

Compute $2A - 3B$ where $A = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -1 \\ 5 & 2 \end{bmatrix}$.

$2A = \begin{bmatrix} 2 & 8 \\ -4 & 6 \end{bmatrix}, \quad 3B = \begin{bmatrix} 0 & -3 \\ 15 & 6 \end{bmatrix}$

$2A - 3B = \begin{bmatrix} 2-0 & 8-(-3) \\ -4-15 & 6-6 \end{bmatrix} = \begin{bmatrix} 2 & 11 \\ -19 & 0 \end{bmatrix}$

Matrix Multiplication

Matrix multiplication is the most important — and most misunderstood — matrix operation. It is not done entry by entry.

When Is Multiplication Defined?

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$.

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

$\underset{m \times n}{A} \cdot \underset{n \times p}{B} = \underset{m \times p}{AB}$

Memory aid: The inner dimensions must match ($n = n$). The outer dimensions give the size of the result ($m \times p$).

A $2 \times 3$ matrix times a $3 \times 4$ matrix gives a $2 \times 4$ matrix. A $2 \times 3$ matrix times a $2 \times 3$ matrix is undefined (3 does not equal 2).

The Row-by-Column Rule

Each entry of the product $AB$ is computed by taking the dot product of a row of $A$ with a column of $B$:

$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$

In plain English: to find the entry in row $i$, column $j$ of the product, multiply each entry in row $i$ of $A$ by the corresponding entry in column $j$ of $B$, then add the results.

Example 5: Multiplying 2x2 Matrices

$AB = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} 5 & -1 \\ 0 & 2 \end{bmatrix}$

Entry $(1,1)$: Row 1 of $A$ dotted with Column 1 of $B$: $2(5) + 3(0) = 10$

Entry $(1,2)$: Row 1 of $A$ dotted with Column 2 of $B$: $2(-1) + 3(2) = 4$

Entry $(2,1)$: Row 2 of $A$ dotted with Column 1 of $B$: $1(5) + 4(0) = 5$

Entry $(2,2)$: Row 2 of $A$ dotted with Column 2 of $B$: $1(-1) + 4(2) = 7$

$AB = \begin{bmatrix} 10 & 4 \\ 5 & 7 \end{bmatrix}$

Example 6: Multiplying Matrices with Different Dimensions

$\begin{bmatrix} 1 & 0 & 2 \\ 3 & 1 & -1 \end{bmatrix} \begin{bmatrix} 4 \\ -2 \\ 5 \end{bmatrix}$

$A$ is $2 \times 3$ and $B$ is $3 \times 1$. Inner dimensions match (3 = 3), so the product is $2 \times 1$.

Entry $(1,1)$: $1(4) + 0(-2) + 2(5) = 4 + 0 + 10 = 14$

Entry $(2,1)$: $3(4) + 1(-2) + (-1)(5) = 12 - 2 - 5 = 5$

$AB = \begin{bmatrix} 14 \\ 5 \end{bmatrix}$

Matrix Multiplication Is NOT Commutative

This is the single most important fact about matrix multiplication: in general, $AB \neq BA$.

Consider $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.

$AB = \begin{bmatrix} 1(0)+2(1) & 1(1)+2(0) \\ 0(0)+1(1) & 0(1)+1(0) \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix}$

$BA = \begin{bmatrix} 0(1)+1(0) & 0(2)+1(1) \\ 1(1)+0(0) & 1(2)+0(1) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 2 \end{bmatrix}$

$AB \neq BA$. In some cases, $AB$ might be defined but $BA$ might not even exist (if the dimensions are incompatible in the reverse order).

The Identity Matrix

The identity matrix $I_n$ is the $n \times n$ matrix with 1s on the diagonal and 0s everywhere else:

$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

The identity matrix is the matrix equivalent of the number 1:

$AI = A \quad \text{and} \quad IA = A$

for any matrix $A$ where the dimensions are compatible.

Example 7: Multiplying by the Identity Matrix

$\begin{bmatrix} 3 & -2 \\ 7 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 3(1)+(-2)(0) & 3(0)+(-2)(1) \\ 7(1)+5(0) & 7(0)+5(1) \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 7 & 5 \end{bmatrix}$

The result is the original matrix — unchanged, as expected.

Properties of Matrix Operations

Real-World Application: Production Cost Analysis

A factory makes two products using three raw materials. The production matrix $P$ shows how many units of each material each product requires, and the cost vector $C$ gives the cost per unit of each material:

$P = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 4 & 2 \end{bmatrix}, \quad C = \begin{bmatrix} 5 \\ 8 \\ 3 \end{bmatrix}$

The total material cost per product is:

$PC = \begin{bmatrix} 3(5) + 2(8) + 1(3) \\ 1(5) + 4(8) + 2(3) \end{bmatrix} = \begin{bmatrix} 34 \\ 43 \end{bmatrix}$

Product 1 costs 34 per unit in materials, and Product 2 costs 43 per unit. This is exactly how manufacturers compute costs across thousands of products and materials — as a single matrix multiplication.

Common Mistakes

1. Multiplying entry by entry. Matrix multiplication uses the row-by-column dot product, not entry-by-entry multiplication. The entry-by-entry product (called the Hadamard product) is a separate operation.
2. Assuming $AB = BA$. Matrix multiplication is not commutative. Always respect the order.
3. Ignoring dimension compatibility. Before multiplying, check that the inner dimensions match. A $2 \times 3$ times a $2 \times 3$ is undefined.
4. Confusing $m \times n$ with $n \times m$. A $2 \times 3$ matrix has 2 rows and 3 columns, not the reverse. Dimensions are always rows-by-columns.

Practice Problems

Key Takeaways

• Matrix dimensions are always stated rows-by-columns: an $m \times n$ matrix has $m$ rows and $n$ columns
• Addition and subtraction require identical dimensions and operate entry by entry
• Scalar multiplication multiplies every entry by the scalar
• Matrix multiplication uses the row-by-column dot product — it is NOT entry by entry
• Multiplication $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$; the result is $m \times p$
• Matrix multiplication is not commutative: $AB \neq BA$ in general
• The identity matrix $I$ satisfies $AI = IA = A$, making it the multiplicative identity for matrices

Return to College Algebra for more topics in this section.