Key takeaways
- det = a(ei − fh) − b(di − fg) + c(dh − eg), expanding along the first row.
- The signs go + − +. The minus on the middle term is the most-missed step.
- Each small part is a 2×2 determinant: cover the row and column of the entry.
- A determinant of 0 means no inverse and no single solution.
In this guide
What is a 3×3 determinant?
A 3×3 matrix has 3 rows and 3 columns, written [a b c; d e f; g h i]. It can stretch, turn and flip shapes in 3D space. Its determinant is one number that tells you how much it scales volume.
Why does the 3×3 determinant matter?
Three equations in three unknowns turn up everywhere, from mixing chemicals to balancing budgets. Before you spend time solving them, the determinant tells you whether there is exactly one answer.
If the determinant is 0, the matrix flattens 3D space into a plane or a line. The system then has no solution or infinitely many, never exactly one.
The formula
Write the matrix as [a b c; d e f; g h i] and expand along the first row:
| Symbol | What it means | Example |
|---|---|---|
| a, b, c | The first row | 2, 0, 1 |
| d, e, f | The second row | 1, 3, 2 |
| g, h, i | The third row | 1, 1, 4 |
| ei − fh | Minor of a (cover row 1, column 1) | 12 − 2 = 10 |
| di − fg | Minor of b (cover row 1, column 2) | 4 − 2 = 2 |
| dh − eg | Minor of c (cover row 1, column 3) | 1 − 3 = −2 |
Each minor is found with the 2×2 determinant rule. For the example, det = 2 × 10 − 0 × 2 + 1 × (−2) = 18.
How to find a 3×3 determinant step by step
- 1Cover row 1 and column 1. Find the 2×2 determinant
ei − fhand multiply it by a. - 2Cover row 1 and column 2. Find
di − fgand multiply it by b. - 3Cover row 1 and column 3. Find
dh − egand multiply it by c. - 4Combine with + − + signs. First term, minus the second, plus the third.
Now enter your own 3×3 matrix in the calculator below and follow the working.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Covering the wrong row or column. Always cover the row and the column of the entry you are using.
- Sign errors inside the minors. Use brackets:
−2 × 7 = −14, and3 − (−4) = 7. - Using the 2×2 diagonal rule on the whole matrix. That only works with the full rule of Sarrus, which uses three diagonals each way.
Tips and tricks
- Triangular matrix? If everything below the main diagonal is 0, the determinant is just a × e × i.
- Two equal rows (or one row a multiple of another) always give a determinant of 0.
- Swapping two rows flips the sign of the determinant.
Where you'll use it in real life
- Algebra: checking whether three equations in three unknowns have exactly one solution.
- Geometry: the volume of a slanted box (parallelepiped) made by three edge vectors.
- Physics and graphics: the same pattern builds the cross product.
- Calculus: the Jacobian that rescales areas and volumes when you change variables.
Quick summary
To find a 3×3 determinant, expand along the first row: multiply each entry by its 2×2 minor and combine with + − + signs. A result of 0 means no inverse and no single solution. Use the calculator above to check your work with every step shown.
Worked example
A system of three equations has the coefficient matrix [2 0 1; 1 3 2; 1 1 4]. Before solving it, you want the determinant to check that the system has exactly one solution.
- 01det = a(ei - fh) - b(di - fg) + c(dh - eg)
- 02ei - fh = 3×4 - 2×1 = 10
- 03di - fg = 1×4 - 2×1 = 2
- 04dh - eg = 1×1 - 3×1 = -2
- 05det = 2(10) - 0(2) + 1(-2)
- 06det = 18
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does a determinant of 0 mean for a 3×3 matrix?
The matrix is singular: it has no inverse, and a system of equations with this matrix does not have exactly one solution.
Can I expand along a different row or column?
Yes. You get the same answer from any row or column, as long as you use the checkerboard sign pattern + − + / − + − / + − +. Choosing a row with zeros saves work.
What is the determinant of the 3×3 identity matrix?
It is 1. The identity matrix leaves every shape unchanged, so volume is scaled by 1.
