Math
Matrix Determinant Calculator
Type a square matrix and get its determinant exactly, with every step written out.
For 2 x 2 and 3 x 3 matrices you see the ad - bc formula or the cofactor expansion along row 1 with each minor computed; for 4 x 4 and larger you see the row operations that turn the matrix triangular. Fractions and decimals stay exact, and the panel tells you whether det = 0 (singular) or the matrix is invertible.
One row per line, entries separated by spaces or commas. The matrix must be square (same number of rows and columns). Fractions like 1/2 stay exact.
Any row or column gives the same determinant. Pick the one with the most zeros to skip terms; the calculator suggests it when a better line exists.
Try an example
Result
Determinant
—
- Decimal value
- —
- Invertible?
- —
- Method
- —
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Study path
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What the determinant tells you
The One number computed from a square matrix that tells you whether it can be inverted. is a single number computed from a square A rectangular grid of numbers arranged in rows and columns.. It is zero exactly when the rows (or columns) are linearly dependent, which is the same as saying the matrix has no inverse. When it is not zero, its How far a number sits from zero, so the answer is never negative. is the factor by which the matrix scales area (2 x 2) or volume (3 x 3), and its sign says whether orientation is preserved (+) or flipped (-).
Determinant formulas
| Sign pattern (-1)^(i+j) | col 1 | col 2 | col 3 |
|---|---|---|---|
| row 1 | + | - | + |
| row 2 | - | + | - |
| row 3 | + | - | + |
How the steps are chosen
| Size | Method shown | Why |
|---|---|---|
| 1 x 1 | The entry itself | det [a] = a |
| 2 x 2 | ad - bc | Fastest formula; every textbook starts here |
| 3 x 3 | Cofactor expansion along the row or column you choose, plus a rule-of-Sarrus check | Shows each 2 x 2 minor and the alternating signs; the diagonal rule confirms the answer |
| 4 x 4 to 8 x 8 | Row reduction to triangular form | Cofactor expansion explodes to n! terms; elimination is what you would do by hand |
How to use it
- Enter the matrix with one row per line. Use spaces or commas between entries.
- For a 3 x 3 matrix, optionally choose which row or column to expand along. The default is row 1; a row or column with zeros is faster and the panel points it out.
- Read det(A) in the header. Fractions are exact; the decimal is shown beside them.
- Open Show the work for the A smaller determinant with a plus or minus sign attached, used to expand a bigger matrix. expansion or the row operations, and use the invertibility status to decide whether an inverse exists.
How to read the answer
- det = 0: Describes a square matrix whose determinant is zero, so it has no inverse. matrix. No inverse, dependent rows, and Ax = b has either no solution or infinitely many.
- det > 0: Describes a matrix that has an inverse, which happens when its determinant is not zero.; the transformation keeps orientation.
- det < 0: invertible; the transformation reflects (flips orientation).
- |det| large: the matrix stretches area or volume a lot; |det| tiny but nonzero: nearly singular.
Useful determinant properties
- det(Aᵀ) = det(A), so expanding along a column works exactly like expanding along a row.
- det(AB) = det(A)·det(B), and det(A⁻¹) = 1 / det(A).
- Swapping two rows changes the sign; multiplying one row by k multiplies the determinant by k; adding a multiple of one row to another leaves it unchanged.
- det(kA) = kⁿ·det(A) for an n x n matrix, which is why the panel also shows det(2A).
- For triangular or diagonal matrices the determinant is just the product of the diagonal entries.
Common mistakes
- Forgetting the alternating signs (+ - +) in a cofactor expansion.
- Trying to take the determinant of a non-square matrix; it is undefined, and the calculator says so.
- Using the 2 x 2 rule ad - bc on a 3 x 3 matrix. The 3 x 3 formula has six products, not two.
- Rounding fractions mid-calculation. Keep 1/3 as 1/3 until the end; this page does that automatically.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
3 x 3 with cofactor expansion
A = [2 1 1; 1 3 2; 1 0 0]
Determinant
-1
2 x 2 quick formula
A = [4 7; 2 6] → 4·6 - 7·2 = 10
Determinant
10
Singular matrix
Row 2 is twice row 1, so det = 0
Determinant
0
Fractions stay exact
The 3 x 3 Hilbert matrix has det = 1/2160
Determinant
1/2160
4 x 4 by row reduction
A = [1 0 2 -1; 3 0 0 5; 2 1 4 -3; 1 0 5 0] → det = 30
Determinant
30
Expand along a column with zeros
A = [1 4 -1; 0 2 -3; 6 11 5], expanding along column 1 skips the middle term → det = -17
Determinant
-17
Upper triangular shortcut
A = [3 7 -2; 0 -1 5; 0 0 4] → 3 × (-1) × 4 = -12
Determinant
-12
Negative determinant (reflection)
A = [0 1; 1 0] swaps the axes, so det = -1
Determinant
-1
Decimal entries
A = [0.5 0.25; 1.5 2] → 1 - 0.375 = 5/8
Determinant
5/8
Not square
A 2 x 3 matrix has no determinant
Determinant
Error
Frequently asked questions
What does a determinant of zero mean?+
A zero determinant means the matrix is singular: its rows are linearly dependent, it has no inverse, and it squashes space onto a lower dimension (a 2 x 2 matrix with det 0 sends the whole plane onto a line or a point).
Can a determinant be negative?+
Yes. The sign records orientation. A negative determinant means the transformation includes a reflection; its absolute value is still the area or volume scale factor.
Which row or column should I expand along?+
Any row or column gives the same answer. Pick the one with the most zeros to skip terms. This calculator always shows row 1 so the steps match the standard textbook layout.
How do you find the determinant of a 4 x 4 matrix?+
Either expand along a row into four 3 x 3 determinants, or row-reduce to an upper triangular matrix and multiply the diagonal, flipping the sign for each row swap. The calculator uses row reduction for 4 x 4 and larger because it is far fewer steps.
Does the determinant exist for a non-square matrix?+
No. Determinants are defined only for square matrices. For a rectangular matrix, look at its rank instead.
Is the determinant of the transpose the same?+
Yes, det(Aᵀ) = det(A) always. That is why column expansion works just like row expansion.
What is the rule of Sarrus?+
A shortcut for 3 x 3 matrices only: copy the first two columns to the right, multiply along the three down-right diagonals and add, multiply along the three up-right diagonals and subtract. The calculator shows this as a check under the cofactor expansion. It does not extend to 4 x 4.
How do I find the determinant of a 3 x 3 matrix by hand?+
Pick a row or column, ideally one with zeros. For each entry, cross out its row and column, find the 2 x 2 determinant that remains (the minor), multiply by the entry and by the sign from the checkerboard pattern, and add the three terms. Choosing Expand along in the calculator shows exactly this for any line.
What is the determinant used for?+
Deciding whether a matrix is invertible (det ≠ 0), solving systems with Cramer's rule, finding eigenvalues from the characteristic polynomial det(A - λI) = 0, computing areas and volumes of transformed shapes, and the Jacobian in change-of-variables integrals.
Does multiplying a matrix by a number multiply the determinant by that number?+
Only for a 1 x 1 matrix. In general det(kA) = kⁿ det(A), because every one of the n rows is scaled by k. Doubling a 3 x 3 matrix multiplies its determinant by 8, which the det(2A) tile shows.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026