Determinant Calculator
Determinant of a square matrix up to 10 × 10 as an exact fraction and a decimal, with the cofactor working shown for 2 × 2 and 3 × 3. How it works
Use the Determinant Calculator
About the Determinant Calculator
The determinant of [1 2; 3 4] is 1×4 − 2×3 = −2, and the determinant of [1 2 3; 4 5 6; 7 8 9] is 0. This determinant calculator works out any square matrix up to 10 × 10 as an exact fraction and writes out the cofactor expansion for 2 × 2 and 3 × 3 matrices. For the inverse or rank of the same matrix, use the Matrix Calculator.
2 × 2 determinants, ad − bc
| Matrix | Working the tool shows | Determinant |
|---|---|---|
| [1 2; 3 4] | ad - bc = 1×4 - 2×3 = -2 | -2 |
| [4 7; 2 6] | ad - bc = 4×6 - 7×2 = 10 | 10 |
| [3 8; 4 6] | ad - bc = 3×6 - 8×4 = -14 | -14 |
| [1 2; 2 4] | ad - bc = 1×4 - 2×2 = 0 | 0 |
| [0.5 1.5; 2 3] | ad - bc = 1/2×3 - 3/2×2 = -3/2 | -3/2 ≈ -1.5 |
| [1/2 1/3; 1/4 1/5] | ad - bc = 1/2×1/5 - 1/3×1/4 = 1/60 | 1/60 ≈ 0.01666666667 |
Rows are separated by semicolons here. In the tool, put each row on its own line.
3 × 3 and 4 × 4 determinants
| Matrix | Working the tool shows | Determinant |
|---|---|---|
| [1 2 3; 4 5 6; 7 8 9] | a(ei - fh) - b(di - fg) + c(dh - eg) = 1×(-3) - 2×(-6) + 3×(-3) = 0 | 0 |
| [2 0 0; 0 3 0; 0 0 4] | a(ei - fh) - b(di - fg) + c(dh - eg) = 2×12 - 0×0 + 0×0 = 24 | 24 |
| [1 2 3; 0 1 4; 5 6 0] | a(ei - fh) - b(di - fg) + c(dh - eg) = 1×(-24) - 2×(-20) + 3×(-5) = 1 | 1 |
| [6 1 1; 4 -2 5; 2 8 7] | a(ei - fh) - b(di - fg) + c(dh - eg) = 6×(-54) - 1×18 + 1×36 = -306 | -306 |
| [2 -3 1; 2 0 -1; 1 4 5] | a(ei - fh) - b(di - fg) + c(dh - eg) = 2×4 - (-3)×11 + 1×8 = 49 | 49 |
| [1 0 2 -1; 3 0 0 5; 2 1 4 -3; 1 0 5 0] | Not shown for 4 × 4 | 30 |
Frequently asked questions
- How do you find the determinant of a 3x3 matrix step by step?
- Expand along the first row: a(ei − fh) − b(di − fg) + c(dh − eg). For [6 1 1; 4 -2 5; 2 8 7], the three 2 × 2 minors are (-2)(7) − 5×8 = -54, 4×7 − 5×2 = 18 and 4×8 − (-2)×2 = 36, so the determinant is 6×(-54) − 1×18 + 1×36 = -306.
- What is the formula for a 2x2 determinant?
- For [a b; c d] it is ad − bc. For [3 8; 4 6] that is 3×6 − 8×4 = 18 − 32 = -14.
- What does a determinant of 0 mean?
- The matrix is singular: it has no inverse, and its rows are linearly dependent. In [1 2 3; 4 5 6; 7 8 9] the third row is twice the second minus the first, so the determinant is 0, and the tool says the matrix has no inverse.
- Can it find the determinant of a 4x4 or larger matrix?
- Yes, up to 10 × 10, by exact Gaussian elimination. [1 0 2 -1; 3 0 0 5; 2 1 4 -3; 1 0 5 0] has determinant 30. The written working is shown for 2 × 2 and 3 × 3 only.
- Can I enter fractions or decimals?
- Yes. Type 1/2 or 0.5 in a cell. [1/2 1/3; 1/4 1/5] gives exactly 1/60, also shown as 0.01666666667.
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