What this calculator does
The determinant is a single number that says whether a matrix can be inverted, and geometrically how much it scales area or volume. A determinant of zero means the transformation collapses space into a lower dimension, which cannot be undone.
That collapse is what singular means. The matrix with rows 1,2,3 then 4,5,6 then 7,8,9 has a determinant of exactly zero, because its rows are not independent: the third is twice the second minus the first.
The formula
A 2x2 determinant is ad minus bc. A 3x3 is computed by cofactor expansion along the first row, which reduces it to three 2x2 determinants.
| Term | Meaning |
|---|---|
| Determinant | A scalar describing how a matrix scales area or volume. |
| Singular | A determinant of zero, meaning no inverse exists. |
| Cofactor expansion | The method for reducing a larger determinant to smaller ones. |
The inputs explained
| Field | What to enter |
|---|---|
| Matrix size | Matrix size. The third row and column entries are ignored for a 2x2. |
| Row 1, Col 1 | Row 1, column 1. |
| Row 1, Col 2 | Row 1, column 2. |
| Row 1, Col 3 (3×3 only) | Row 1, column 3, used for a 3x3 only. |
| Row 2, Col 1 | Row 2, column 1. |
| Row 2, Col 2 | Row 2, column 2. |
| Row 2, Col 3 (3×3 only) | Row 2, column 3, used for a 3x3 only. |
| Row 3, Col 1 (3×3 only) | Row 3, column 1, used for a 3x3 only. |
| Row 3, Col 2 (3×3 only) | Row 3, column 2, used for a 3x3 only. |
| Row 3, Col 3 (3×3 only) | Row 3, column 3, used for a 3x3 only. |
When to use it
Checking whether a matrix is invertible
A zero determinant means no inverse exists, which is the first thing to test.
Solving a linear system
Cramer's rule uses determinants directly, and a zero one means no unique solution.
Finding a scaling factor
The absolute determinant is how much the transformation multiplies area or volume.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Where does the determinant reach zero?
The same matrix with one entry varied.
Questions
What does a determinant of zero mean geometrically?
That the transformation flattens space. A 2x2 matrix with zero determinant maps the plane onto a line, and a 3x3 maps space onto a plane or line. Information is lost, which is why the operation cannot be reversed.
What does a negative determinant mean?
That the transformation reverses orientation, flipping the space as well as scaling it. The magnitude still gives the scaling factor; the sign tells you whether handedness was preserved.
How are larger determinants calculated?
Cofactor expansion works at any size but becomes impractical quickly, since the work grows as the factorial of the dimension. Practical computation uses row reduction to triangular form, then multiplies the diagonal.
Why is this matrix singular?
Because its rows are linearly dependent: one row can be built from the others. That dependence means the rows do not span the full space, which is exactly what a zero determinant detects.
For inverting a 2x2 matrix, see the matrix inverse calculator. For eigenvalues, see the eigenvalues calculator.