# What is the determinant of my matrix?

Finds the determinant of a square matrix from 1 × 1 to 6 × 6 by Gaussian elimination in exact fractions, with every row operation and whether the matrix is invertible.

- Page: https://www.acalculator.org/math/determinant-calculator
- JSON spec: https://www.acalculator.org/math/determinant-calculator.json
- Version: c807e55d9efa

## Default answer

Example with the default inputs (Matrix A [2, -3, 1; 2, 0, -1; 1, 4, 5]): The determinant of the matrix is 49.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Matrix A | The square matrix, 1 to 6 rows and the same number of columns. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| determinant | Determinant det(A) | The determinant as an exact fraction in lowest terms. |
| decimal | Determinant in decimals | The determinant rounded half up to 10 significant figures. |
| invertible | Does A have an inverse? | Yes when the determinant is not 0. |
| steps | Elimination | The row operations to upper triangular form, the triangular matrix, and the product of its diagonal. |

## Method

Gaussian elimination to upper triangular form; det(A) = (−1)^(row swaps) × the product of the diagonal. For 2 × 2, det = ad − bc.

## Assumptions

- Every cell is read as an exact fraction (1/3 is 1/3, 0.1 is 1/10), so the determinant has no rounding error.
- The pivot for each column is the first row, from the diagonal down, with a non-zero entry.
- The matrix is square, 1 × 1 to 6 × 6.

## Worked examples

1. a = 2 or -3 or 2 or 0 gives determinant = 49, decimal = 49, invertible = yes, steps = R2 → R2 − R1; R3 → R3 − 1/2 × R1; R3 → R3 − 11/6 × R2; Upper triangular: [2, -3, 1; 0, 3, -2; 0, 0, 49/6]; det = 2 × 3 × 49/6 = 49. Source: OpenStax, Precalculus 2e, §9.8, 3 × 3 determinants (https://openstax.org/books/precalculus-2e/pages/9-8-solving-systems-with-cramers-rule).
2. a = 4 or 7 or 2 or 6 gives determinant = 10, invertible = yes.
3. a = 0 or 1 or 1 or 0 gives determinant = -1, steps = R1 ↔ R2; Upper triangular: [1, 0; 0, 1]; det = (-1)^1 × 1 × 1 = -1.
4. a = 1 or 2 or 4 or 5 gives determinant = 0, invertible = no.
5. a = 0.5 or 0.333333 or 0.25 or 1 gives determinant = 5/12, decimal = 0.4166666667.
6. a = -7 or undefined or undefined gives determinant = -7, steps = Upper triangular: [-7]; det = -7.

## FAQ

### How do I find the determinant of a 2 × 2 matrix?

For [a, b; c, d], det = ad − bc. For [4, 7; 2, 6], det = 4 × 6 − 7 × 2 = 24 − 14 = 10.

### How do I find the determinant of a 3 × 3 matrix?

Expand along the first row: det = a(ei − fh) − b(di − fg) + c(dh − eg) for [a, b, c; d, e, f; g, h, i]. For [2, −3, 1; 2, 0, −1; 1, 4, 5]: 2 × 4 − (−3) × 11 + 1 × 8 = 49. The calculator uses row elimination instead, which gives the same answer and works for larger matrices.

### What does a determinant of 0 mean?

The matrix is singular: its rows (and columns) are dependent, it has no inverse, and a system of equations with it has either no solution or infinitely many. [1, 2, 3; 4, 5, 6; 7, 8, 9] has determinant 0 because row 3 − row 2 equals row 2 − row 1.

### How do row operations change the determinant?

Swapping two rows flips its sign. Adding a multiple of one row to another leaves it unchanged. Multiplying a row by k multiplies it by k. The calculator only swaps rows and adds multiples, so the determinant of the triangular result, times −1 for each swap, is the answer.

### What does the determinant tell me?

Its size is how much the matrix scales area (2 × 2) or volume (3 × 3), and its sign tells whether it flips orientation. A determinant of −1 (as for [0, 1; 1, 0]) keeps areas the same size but mirrors them.

### Can I use fractions or decimals?

Yes. Type 1/3 or 0.25 in any cell. The calculator reads 1/3 as exactly one third and 0.1 as one tenth, and works in exact fractions, so the determinant has no rounding error. [1/2, 1/3; 1/4, 1] has determinant 5/12.

## Sources

- Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), §5.1 The Properties of Determinants and §5.2 Permutations and Cofactors.
- OpenStax, Precalculus 2e, §9.8 Solving Systems with Cramer’s Rule (2 × 2 and 3 × 3 determinants). https://openstax.org/books/precalculus-2e/pages/9-8-solving-systems-with-cramers-rule