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.

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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. For the wider topic that pairs matrices with their determinants, see matrices and determinants.

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

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, 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.