Determinant Calculator: 2×2, 3×3, 4×4 and 5×5 Matrix with Steps
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The determinant of the matrix with rows (2, −1, 3), (0, 4, 1) and (5, 2, −2) is −85, so it is invertible.
The determinant of this 3 × 3 matrix is −85. Because it is not 0, the matrix is invertible.
Your matrix
| Column 1 | Column 2 | Column 3 |
|---|---|---|
| 2 | −1 | 3 |
| 0 | 4 | 1 |
| 5 | 2 | −2 |
Show the steps
- Expand along the first row: det = a(ei − fh) − b(di − fg) + c(dh − eg).
- First term: 2 × (-10) = -20.
- Second term: −(-1) × (-5) = -5.
- Third term: 3 × (-20) = -60.
- Add the three terms: -85.
Determinant
2 × 2: det = ad − bc
3 × 3: det = a(ei − fh) − b(di − fg) + c(dh − eg)
Larger: reduce to triangular form, multiply the diagonal, flip the sign for each row swap
- A square matrix is invertible exactly when its determinant is not zero.
- Fractions and decimals work. Decimals can have up to six places.
Estimates for planning and learning, not financial, tax, legal or medical advice. Check figures with your lender, tax preparer or doctor before you act.
About this calculator
The determinant is a single number computed from a square matrix. It tells you whether the matrix has an inverse, whether a system of linear equations has a unique solution, and how a transformation scales area or volume. This calculator works with exact fractions, so you get a clean answer instead of a rounded decimal.
Pick the size from 2 × 2 to 5 × 5, type one row per line, and the page shows the determinant, whether the matrix is invertible, its rank and the steps. For 2 × 2 and 3 × 3 matrices you get the textbook formula. For larger matrices you see the row operations.
How to use it
- Choose the matrix size.
- Type the matrix, one row per line, with the numbers separated by spaces or commas. Fractions such as 1/2 are fine.
- Read the determinant, and open Show the steps for the working.
The 2 × 2 and 3 × 3 formulas
For a 2 × 2 matrix with rows (a, b) and (c, d), the determinant is ad − bc. For a 3 × 3 matrix with rows (a, b, c), (d, e, f) and (g, h, i), expand along the first row: a(ei − fh) − b(di − fg) + c(dh − eg).
Row reduction
For bigger matrices, use row operations to reach a triangular matrix. Adding a multiple of one row to another does not change the determinant. Swapping two rows flips its sign. Multiplying a row by a number multiplies the determinant by that number. At the end, the determinant is the product of the diagonal entries.
What a zero determinant means
If the determinant is 0, the matrix is singular: it has no inverse, its rows are linearly dependent, and a system with this coefficient matrix has either no solution or infinitely many.
Sources
Quick check: test yourself
What is the determinant of [[3, 2], [1, 4]]?
Correct answer: 10. ad − bc = 3 × 4 − 2 × 1 = 10.
A matrix has determinant 0. What does that mean?
Correct answer: It has no inverse. A square matrix is invertible only when its determinant is not zero.
What happens to the determinant when you swap two rows?
Correct answer: It changes sign. A row swap multiplies the determinant by −1.
Pick an answer for each question.
Common questions
How do I find the determinant of a 2×2 matrix?
Multiply the top left and bottom right, then subtract the product of the top right and bottom left: ad − bc.
What does a determinant of 0 mean?
The matrix is singular. It has no inverse, and its rows or columns are dependent.
Can the determinant be negative?
Yes. The sign tells you whether the transformation keeps or flips orientation.
How do I find a 3×3 determinant?
Expand along a row or column using cofactors, or use row reduction. The calculator shows the first-row expansion.
Does the determinant of the transpose change?
No. A matrix and its transpose have the same determinant.
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