What a matrix is
A matrix is a rectangular array of numbers. It helps us package information in rows and columns so we can work with it more systematically.
Definition
Matrix
A matrix is a rectangular array of numbers arranged in rows and columns.
Read the size first
The size of a matrix is written as m × n.
mis the number of rows.nis the number of columns.
If m = n, the matrix is square.
Look at entries one by one
If A is a matrix, then a_{ij} means the entry in row i and column j.
That index notation matters later, because it tells you exactly where a number
lives inside the matrix.
Worked example
Reading a matrix carefully
Let
This matrix has 2 rows and 3 columns, so its size is 2 × 3.
Its entry in row 2, column 3 is 4.
Why this is useful
Once you know how to read a matrix, you can use it to store a system of equations, track row operations, and later test whether a matrix is invertible.
Use the small explorer below as you read. Change one entry, pick one output cell, and watch how one row of the first matrix and one column of the second matrix produce that single entry.
Try it here
Matrix multiplication visualizer
The live widget updates each entry of AB as you change the entries of A and B.
Result
| 8 | 9 |
| 3 | 4 |
8 = 1×2 + 2×3
Common mistake
Common mistake
Do not swap rows and columns
The first index is the row number, not the column number. a_{23} means row 2,
column 3.
Quick check
Quick check
What is the size of a matrix with 3 rows and 4 columns?
Solution
Answer
Prerequisite link
If you want to see how a system becomes a matrix, review 1.1 Equations and solution sets.