# Determinants — Definition, Formula, and Examples

TL;DR

A determinant is a single number computed from a square matrix that tells you whether the matrix can be inverted and how it scales area or volume. This article defines determinants, works the 2×2 and 3×3 expansions step by step, lists the properties that speed the arithmetic up, and shows where the answer matters.

BT

Last updated on July 19, 2026. 10 min read.

## What a determinant is

A **determinant** is a scalar (a single number) assigned to a square matrix — a grid with the same number of rows and columns. It is written \(\text{det}(A)\) or with vertical bars, \(|A|\). A matrix that is not square has no determinant.

The determinant answers two questions at once. It tells you whether the matrix is **invertible**: a matrix can be inverted exactly when its determinant is not zero. It also measures how the matrix stretches or shrinks space — in two dimensions, the absolute value of the determinant is the area of the parallelogram formed by the matrix's columns.

This article is about the determinant as a concept and how to expand it. For the mechanics of one worked grid end to end, see the companion article on the [determinant of a matrix](/content/math/algebra/determinant-of-matrix/index.html). For the wider topic that pairs matrices with their determinants, see [matrices and determinants](/content/math/algebra/matrices-and-determinants/index.html).

### How is a determinant different from the matrix itself?

A matrix is the whole grid of numbers. The determinant is one number squeezed out of that grid. You can add and multiply matrices; the determinant is just a readout — like a temperature taken from a whole room.

## The 2×2 formula

For a 2×2 matrix

\[ A = \begin{bmatrix} a & b \\
 c & d \end{bmatrix} \]

the determinant is the product of the main diagonal minus the product of the other diagonal:

\[ \text{det}(A) = ad - bc \]

## The 3×3 formula (cofactor expansion)

For a 3×3 matrix

\[ A = \begin{bmatrix} a & b & c \\
 d & e & f \\
 g & h & i \end{bmatrix} \]

you expand along the top row, attaching the sign pattern +,−,+,−,+,−,+ and multiplying each entry by the 2×2 **minor** left when you delete that entry's row and column:

\[ \text{det}(A) = a\begin{vmatrix} e & f \\
 h & i \end{vmatrix} - b\begin{vmatrix} d & f \\
 g & i \end{vmatrix} + c\begin{vmatrix} d & e \\
 g & h \end{vmatrix} \]

A **minor** is the smaller determinant left after deleting one row and one column. A **cofactor** is that minor with its sign attached from the checkerboard pattern below.

\[ \begin{bmatrix} + & - & + \\
 - & + & - \\
 + & - & + \end{bmatrix} \]

The sign in position (row i, column j) is \((-1)^{i+j}\).

## Variable Key

| Symbol          | Meaning                                                                 |  
|------------------|-------------------------------------------------------------------------|  
| a, b, c, d      | The four entries of a 2×2 matrix, row by row                          |  
| a through i     | The nine entries of a 3×3 matrix, row by row                           |  
| \text{det}(A)  | The determinant of matrix A, a single scalar                          |  
| Minor           | The 2×2 determinant left after deleting one entry's row and column     |  
| Cofactor        | A minor with the checkerboard sign attached                            |

## Examples of Determinants

### Example 1

**Find the determinant of** \[ \begin{bmatrix} 4 & 3 \\
 2 & 5 \end{bmatrix} \].

\[ \text{det} = (4)(5) - (3)(2) \]

\[ \text{det} = 20 - 6 \]

\[ \text{det} = 14 \]

The result is nonzero, so this matrix is invertible.

### Example 2

**Find the determinant of** \[ \begin{bmatrix} 7 & 2 \\
 -3 & 5 \end{bmatrix} \], watching the sign on the negative entry.

\[ \text{det} = (7)(5) - (2)(-3) \]

\[ \text{det} = 35 + 6 \]

\[ \text{det} = 41 \]

### Example 3

**Expand the determinant of** \[ \begin{bmatrix} 1 & 2 & 3 \\
 4 & 5 & 6 \\
 7 & 8 & 10 \end{bmatrix} \] along the top row.

\[ \text{det} = 1\begin{vmatrix} 5 & 6 \\
 8 & 10 \end{vmatrix} - 2\begin{vmatrix} 4 & 6 \\
 7 & 10 \end{vmatrix} + 3\begin{vmatrix} 4 & 5 \\
 7 & 8 \end{vmatrix} \]

Calculating the minors gives:

First minor:

\[ (5)(10) - (6)(8) = 2 \]

Second minor:

\[ (4)(10) - (6)(7) = -2 \]

Third minor:

\[ (4)(8) - (5)(7) = -3 \]

Now assemble with the signs:

\[ \text{det} = 1(2) - 2(-2) + 3(-3) \]

\[ \text{det} = 2 + 4 - 9 \]

\[ \text{det} = -3 \]

### Example 4

**Find the determinant of** \[ \begin{bmatrix} 2 & 0 & 1 \\
 3 & 0 & 4 \\
 5 & 0 & 6 \end{bmatrix} \].

The middle column is all zeros.

\[ \text{det} = 0 \]

### Example 5

**Find the determinant of the upper-triangular matrix** \[ \begin{bmatrix} 3 & 7 & 2 \\
 0 & 4 & 5 \\
 0 & 0 & 6 \end{bmatrix} \].

\[ \text{det} = 3 \times 4 \times 6 = 72 \]

### Example 6

**Use Cramer's rule to solve the system** \(2x + y = 5 \) **and** \(x + 3y = 10\.**

Write the coefficient matrix and its determinant:

\[ D = \begin{vmatrix} 2 & 1 \\
 1 & 3 \end{vmatrix} = (2)(3) - (1)(1) = 5 \]

Replace the x-column with the constants for \(D_x\):

\[ D_x = \begin{vmatrix} 5 & 1 \\
 10 & 3 \end{vmatrix} = (5)(3) - (1)(10) = 5 \]

Replace the y-column for \(D_y\):

\[ D_y = \begin{vmatrix} 2 & 5 \\
 1 & 10 \end{vmatrix} = (2)(10) - (5)(1) = 15 \]

Then:

\[ x = \frac{D_x}{D} = \frac{5}{5} = 1 \]

\[ y = \frac{D_y}{D} = \frac{15}{5} = 3 \]

Because \(D \neq 0\), the system has exactly one solution.

## Why "det" is Cramer's yes-or-no light

The determinant earns its keep as a single yes-or-no signal. Before you spend effort inverting a matrix or solving a system, one number tells you whether the work is even possible.

- **Invertibility gate.** A  [singular matrix](/content/math/algebra/singular-matrix/index.html), one with \(\text{det} = 0\), cannot be inverted.

- **Geometry of transformations.** The determinant is the area scale factor in 2D and the volume scale factor in 3D.
 
## Properties That Shrink The Arithmetic

A few properties let you avoid full expansion. Each one is a legal move that either leaves the determinant unchanged or changes it in a predictable way.

- **Row/column swap flips the sign.**
- **Two equal rows force zero.**
- **Adding a multiple of one row to another changes nothing.**
- **Triangular shortcut.**
- **Transpose is invariant.**
- **Product rule.**

## Common Mistakes With Determinants

### Mistake 1: Dropping the checkerboard sign

### Mistake 2: Deleting the wrong row or column for a minor

### Mistake 3: Treating the inner minus as if it weren't there

## Conclusion

A **determinant** is a single number computed from a square matrix; it is zero exactly when the matrix cannot be inverted. The 2×2 determinant is \(ad-bc\); the 3×3 expands by cofactors along a row.

## A Practical Next Step

Practice the 2×2 formula until the sign handling feels automatic. To take determinants further with a teacher, explore Bhanzu's [algebra tutor](/content/math/tutor/algebra/index.html).