Determinant Calculator
Type the numbers of the matrix.
Determinant:
Is it invertible?:
Area of the parallelogram:
Orientation of the columns:
Calculation:
This calculator finds the determinant of a 2x2 or 3x3 matrix and shows the work step by step. For 3x3 matrices you can choose the rule of Sarrus or cofactor expansion along any row or column, and the calculator checks the answer with the other method. For 2x2 it draws the parallelogram made by the columns, whose signed area is the determinant.
Further down you will find the formulas, worked examples, a table of special cases and the most common questions.
How to use the determinant calculator?
Step 1: Choose the size of the matrix: 2x2 or 3x3. For 3x3, also pick the method.
Step 2: Type the numbers of the matrix. A decimal point or comma both work, for example 2.5 or 2,5, and negative numbers are fine. In 2x2 you can also drag the tips of the arrows.
Step 3: Press Calculate or the Enter key. The result updates as you type.
Step 4: Read the determinant, whether the matrix is invertible, and the area or volume. The Calculation box shows every operation.
2x2 determinant formula
For a matrix with rows (a, b) and (c, d), the determinant is the product of the main diagonal minus the product of the other diagonal:
| det = a·d - b·c |
If you draw the columns (a, c) and (b, d) as arrows from the origin, the determinant is the signed area of the parallelogram they make: positive when the second arrow is to the left of the first (counterclockwise) and negative when it is to the right. When the determinant is 0 the parallelogram collapses into a line.
3x3 determinant: rule of Sarrus
The rule of Sarrus only works for 3x3 matrices. Copy the first two columns to the right, add the products of the three diagonals that go down and subtract the products of the three that go up:
| det = aei + bfg + cdh - ceg - afh - bdi |
3x3 determinant: cofactor expansion
Cofactor expansion works for any size. Pick a row or column and add each entry times its sign times the determinant of the 2x2 matrix left after deleting its row and column (the minor). The sign is (-1)i+j, a checkerboard of + and -. It pays to choose the row or column with the most zeros:
| det = a11M11 - a12M12 + a13M13 (along row 1) |
In 3x3 the absolute value of the determinant is the volume of the parallelepiped made by the three columns.
Examples of determinants
Example 1: 2x2 matrix [[4, 3], [2, 5]]
det = 4 × 5 - 3 × 2 = 20 - 6 = 14
The determinant is positive, so the columns turn counterclockwise and the area of the parallelogram is 14.
Example 2: A negative determinant
det = 1 × 3 - 4 × 2 = 3 - 8 = -5
The negative sign means clockwise. The area is still 5, because area has no sign.
Example 3: 3x3 matrix with the rule of Sarrus
Find the determinant of [[2, 1, 3], [0, 4, 1], [5, 2, 1]].
(+) 2 × 4 × 1 + 1 × 1 × 5 + 3 × 0 × 2 = 8 + 5 + 0 = 13
(-) 3 × 4 × 5 + 2 × 1 × 2 + 1 × 0 × 1 = 60 + 4 + 0 = 64
det = 13 - 64 = -51
Cofactors along row 1 give the same value: -51.
Example 4: 3x3 matrix with cofactors
Find the determinant of [[3, 1, 0], [1, 2, 1], [0, 1, 4]] along row 1. Since a13 = 0, that term vanishes.
a11: 3 × (+1) × (2 × 4 - 1 × 1) = 3 × (+1) × 7 = 21
a12: 1 × (-1) × (1 × 4 - 1 × 0) = 1 × (-1) × 4 = -4
a13: 0 × (+1) × (1 × 1 - 2 × 0) = 0 × (+1) × 1 = 0
det = 21 + (-4) + 0 = 17
Table of special cases
| Matrix | Determinant | Why |
|---|---|---|
| [[1, 0], [0, 1]] | 1 | Identity matrix |
| [[2, 5], [0, 3]] | 6 | Triangular: product of the diagonal |
| [[1, 2], [2, 4]] | 0 | Proportional rows |
| [[1, 2], [3, 4]] | -2 | Original |
| [[3, 4], [1, 2]] | 2 | With the rows swapped, the sign flips |
| [[1, 2], [0, 0]] | 0 | A row of zeros |
Frequently asked questions
How do you find the determinant of a 2x2 matrix?
Multiply the main diagonal and subtract the product of the other diagonal. For [[1, 2], [3, 4]]: 1 × 4 - 2 × 3 = -2.
What is the difference between Sarrus and cofactors?
Sarrus is a visual shortcut that only works for 3x3. Cofactors work for any size and are easier when a row or column has zeros. Both give the same result.
What does a determinant of 0 mean?
The matrix is singular: it has no inverse, and its rows or columns depend on each other. For example, [[2, 4], [1, 2]] has determinant 0 because the first row is twice the second. In the drawing, the parallelogram collapses.
How are the determinant and area related?
For a 2x2 matrix, the absolute value of the determinant is the area of the parallelogram made by the columns. For [[3, 0], [0, 5]] it is a 3 by 5 rectangle and the area is 15.
Which row or column should I pick for cofactors?
The one with the most zeros, because every zero removes a term. If there are no zeros, any choice works and the result is the same.
Learn more
- Determinant of a 2x2 matrix: examples with answers
- Determinant of a 3x3 matrix: examples with answers
- Inverse of a 2x2 matrix: examples with answers
- Area of a parallelogram: formulas and examples
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