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Matrix — Definition, Examples & Properties

Key Formula

A=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}
Where:
  • AA = The name of the matrix (typically an uppercase letter)
  • aija_{ij} = The element in row i and column j
  • mm = The number of rows
  • nn = The number of columns

Worked Example

Problem: Given matrices A and B, find A + B where A = [[2, 5], [1, 3]] and B = [[4, -1], [0, 6]].
Step 1: Write out both matrices. Both are 2×2, so addition is defined.
A=[2513],B=[4−106]A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & -1 \\ 0 & 6 \end{bmatrix}
Step 2: Add corresponding elements. Each entry in the result is the sum of the entries in the same position from A and B.
A+B=[2+45+(−1)1+03+6]A + B = \begin{bmatrix} 2+4 & 5+(-1) \\ 1+0 & 3+6 \end{bmatrix}
Step 3: Simplify each entry.
A+B=[6419]A + B = \begin{bmatrix} 6 & 4 \\ 1 & 9 \end{bmatrix}
Answer: A + B = [[6, 4], [1, 9]]

Another Example

This example demonstrates matrix multiplication, which is more involved than addition. Unlike addition, multiplication requires a dot-product process across rows and columns, and the order of multiplication matters (AB ≠ BA in general).

Problem: Multiply matrix A by matrix B where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
Step 1: Check dimensions. A is 2×2 and B is 2×2. The number of columns in A (2) equals the number of rows in B (2), so multiplication is defined. The result will be a 2×2 matrix.
Step 2: Find the entry in row 1, column 1. Multiply each element of row 1 of A by the corresponding element of column 1 of B, then add.
(1)(5)+(2)(7)=5+14=19(1)(5) + (2)(7) = 5 + 14 = 19
Step 3: Find the entry in row 1, column 2.
(1)(6)+(2)(8)=6+16=22(1)(6) + (2)(8) = 6 + 16 = 22
Step 4: Find the entries in row 2 using the same method.
(3)(5)+(4)(7)=43,(3)(6)+(4)(8)=50(3)(5) + (4)(7) = 43, \quad (3)(6) + (4)(8) = 50
Step 5: Write the resulting matrix.
AB=[19224350]AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}
Answer: AB = [[19, 22], [43, 50]]

Frequently Asked Questions

What is the difference between a matrix and a determinant?
A matrix is an array of numbers arranged in rows and columns — it is a collection of data. A determinant is a single number computed from a square matrix. You write a matrix with brackets [ ] and a determinant with vertical bars | |. Every square matrix has a determinant, but a determinant is not a matrix.
What does the size (or order) of a matrix mean?
The size of a matrix is described by its dimensions: m × n, where m is the number of rows and n is the number of columns. For example, a matrix with 3 rows and 2 columns is a 3×2 matrix. When a matrix has the same number of rows and columns, it is called a square matrix.
When do you use matrices in math?
Matrices are used to solve systems of linear equations, perform geometric transformations (like rotations and reflections), and represent data in organized form. They appear throughout algebra, precalculus, linear algebra, computer graphics, physics, and statistics.

Matrix vs. Determinant

MatrixDeterminant
What it isA rectangular array of numbersA single numerical value computed from a square matrix
NotationBrackets or parentheses: [ ] or ( )Vertical bars: | |
Shape requirementCan be any m × n dimensionsOnly defined for square matrices (n × n)
Result typeAn organized collection of numbersA single scalar number
Primary useOrganizing and transforming data, solving systemsTesting invertibility, finding area/volume

Why It Matters

Matrices appear in algebra courses when you solve systems of equations using methods like row reduction or Cramer's rule. They are essential in precalculus and linear algebra for understanding transformations, and they form the backbone of computer graphics, machine learning, and engineering applications. Mastering basic matrix operations — addition, scalar multiplication, and matrix multiplication — is a prerequisite for nearly all advanced mathematics courses.

Common Mistakes

Mistake: Adding or subtracting matrices of different sizes.
Correction: Matrix addition and subtraction require both matrices to have the exact same dimensions. A 2×3 matrix cannot be added to a 3×2 matrix.
Mistake: Assuming matrix multiplication is commutative (that AB = BA).
Correction: Matrix multiplication is generally not commutative. The product AB can be completely different from BA, and in many cases one product may be defined while the other is not. Always multiply in the stated order.

Related Terms