Key takeaways
- For [a b; c d], the determinant is ad − bc.
- Main diagonal product minus the other diagonal product, in that order.
- The determinant is the area scale factor. A negative sign means the shape is flipped.
- If the determinant is 0, the matrix has no inverse.
In this guide
What is a determinant?
A matrix is a grid of numbers in rows and columns. A 2×2 matrix has 2 rows and 2 columns, written [a b; c d]. Its determinant is a single number made from those four entries.
In pictures, a 2×2 matrix stretches, turns and shears shapes on a flat plane. The determinant tells you how much it scales area. A determinant of 3 makes every shape 3 times bigger.
Why does the determinant matter?
The determinant is a quick health check for a matrix. If it is not 0, the matrix can be reversed (it has an inverse), and a system of two equations built from it has exactly one solution. If it is 0, something has collapsed.
A determinant of 0 means the matrix squashes the whole plane onto a line or a point. Once that happens, there is no way to get the original shape back.
A negative determinant is fine. It means the matrix also flips shapes over, like a mirror.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a | Top-left entry | 5 |
| b | Top-right entry | 2 |
| c | Bottom-left entry | 3 |
| d | Bottom-right entry | 4 |
| det(A) | The determinant | 5 × 4 − 2 × 3 = 14 |
Read it as: multiply down the main diagonal, then subtract the product of the other diagonal.
How to find a 2×2 determinant step by step
- 1Write the matrix as [a b; c d]. Label each entry.
- 2Multiply the main diagonal. Work out
a × d. - 3Multiply the other diagonal. Work out
b × c. - 4Subtract.
(a × d) − (b × c)is the determinant.
Now enter your own four entries in the calculator below and check your answer.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Sign errors with negatives. If b = −2 and c = 3, then bc = −6, and ad − (−6) means adding 6.
- Mixing up b and c. It does not change the determinant, but it does matter for the inverse, so keep positions clear.
- Using it on non-square matrices. Only square matrices (2×2, 3×3 and so on) have determinants.
Tips and tricks
- Swapping two rows changes the sign of the determinant.
- For a product of matrices, det(AB) = det(A) × det(B). Try the matrix multiplication calculator to check.
- Next step up: the 3×3 determinant calculator breaks a 3×3 into three of these 2×2 determinants.
Where you'll use it in real life
- Algebra: checking whether simultaneous equations have exactly one solution, and solving them with Cramer's rule.
- Matrices: the first step of the 2×2 inverse.
- Geometry: the area of a parallelogram made by two arrows is the size of their determinant.
- Computer graphics: checking whether a transformation flips or squashes an image.
Quick summary
The determinant of [a b; c d] is ad − bc. It tells you how the matrix scales area, and a value of 0 means there is no inverse. Use the calculator above to find any 2×2 determinant with every step shown.
Worked example
A computer graphics program stretches images using the matrix [5 2; 3 4]. You want to know by what factor it changes the area of a picture.
- 01det(A) = ad - bc
- 02= (5)(4) - (2)(3)
- 03= 20 - 6
- 04= 14
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does it mean if the determinant is 0?
The matrix squashes the plane onto a line or a point. It has no inverse, and the equations it represents do not have exactly one solution.
Can a determinant be negative?
Yes. A negative determinant means the matrix flips shapes over, like a mirror, as well as scaling their area.
What is the determinant of [1 2; 3 4]?
1 × 4 − 2 × 3 = 4 − 6 = −2.
