How do I calculate a 2×2 determinant?

For A = [[a,b],[c,d]], calculate ad − bc. For [[2,1],[1,3]], that gives 6 − 1 = 5.

Can I enter a 3×3 or larger matrix?

Yes, you can request an AI explanation for a square matrix, including cofactor expansion or elimination. Larger inputs are harder to audit; this page does not promise an exact-arithmetic engine or a tested maximum size.

What does a zero determinant mean?

The square matrix is singular and has no inverse. Its columns are linearly dependent. Ax = b may have no solution or infinitely many solutions, depending on b.

Can the determinant be negative?

Yes. Its sign indicates orientation; its absolute value gives area scaling in 2D or volume scaling in 3D.

Can I upload a photo and get steps?

Yes. Upload a clear matrix image and request the method you need. Check the recognized entries and arithmetic because the response is AI-generated.

Matrices, methods, and checks

Determinant Calculator

Use this determinant calculator for AI-guided 2×2 and 3×3 worked solutions, cofactor expansion, and row-operation checks. Enter a square matrix and interpret whether it is invertible.

Type or upload a matrixWorked explanationsNo sign-up required

Enter a problem

determinant calculator

det(A) = ad − bc

2×2 determinants

3×3 cofactor expansion

Larger matrices and row operations

determinant calculator

Work through your determinant problem

Paste each matrix row in brackets, or upload a clear image. State the operation and ask for steps. Responses are AI-generated; check important arithmetic independently.

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determinant calculator

Find the determinant, then understand what it means

The determinant is a scalar defined for square matrices. Match the method to the size and structure of your matrix.

01ad − bc

2×2 determinants

Use ad − bc. A negative result is valid; it describes a reversal of orientation.

02+ − +

3×3 cofactor expansion

Expand across a row or column, showing the minors and the alternating signs. Sarrus works only for 3×3 matrices.

03det(U) = ∏uᵢᵢ

Larger matrices and row operations

Ask for elimination or expansion along a row with zeros. Track swaps and scaling before multiplying the diagonal.

04det(A) = 0 ⇔ singular

Singularity and geometry

det(A) = 0 means a square matrix is singular. The absolute determinant measures area or volume scaling.

Method

How to use the determinant calculator

Keep the original matrix visible while you follow the explanation.

1
Enter a square matrix
A ∈ ℝⁿˣⁿ
Include equal numbers of rows and columns, preserving all signs and fractions.
2
Choose an approach
Cᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ
Ask for ad − bc, cofactor expansion, or elimination as appropriate.
3
Track signs and factors
Rᵢ ↔ Rⱼ ⇒ −det(A)
A row swap changes the sign; scaling one row scales the determinant.
4
Interpret and check
det(A) ≠ 0 ⇒ A⁻¹
Test invertibility or confirm with an alternative expansion or triangular form.

Tip: a zero determinant does not mean the matrix is all zeros. Dependent rows are enough.

Quick reference

Determinant rules worth keeping beside your work

These rules help audit a calculation, especially when a solution uses row reduction.

2×2 formula

det([[a,b],[c,d]]) = ad − bc

Multiply the main diagonal, then subtract the other diagonal product.

Cofactor expansion

det(A) = Σⱼ aᵢⱼ(−1)ⁱ⁺ʲMᵢⱼ

Mᵢⱼ is the determinant after deleting row i and column j; signs use 1-based indices.

Row replacement

Rᵢ ← Rᵢ + kRⱼ

Adding a multiple of a different row leaves the determinant unchanged.

Triangular matrix

det(U) = u₁₁u₂₂…uₙₙ

Undo any row-scaling factors and row-swap signs introduced during elimination.

Worked example

Worked determinant examples

These reviewed examples show the method, result, and an independent check. Use the example prompts above to try them in the solver.

Worked example 1

det([[2,1],[1,3]])

A 2×2 example with a nonzero determinant.

Identify a = 2, b = 1, c = 1, d = 3.
Compute ad − bc = 2·3 − 1·1 = 6 − 1.

Result: det(A) = 5; A is invertible.

Check: Replace R₂ with R₂ − (1/2)R₁: the triangular diagonal is 2, 5/2, whose product is 5.

Worked example 2

det([[1,2,0],[0,3,4],[5,0,6]])

Expand along the first row to see each cofactor sign.

The first-row signs are +, −, +.
The first minor is det([[3,4],[0,6]]) = 18. The second is det([[0,4],[5,6]]) = −20.
Compute 1·18 − 2·(−20) + 0 = 58.

Result: det(A) = 58

Check: R₃ ← R₃ − 5R₁ gives [0,−10,6]; then R₃ ← R₃ + (10/3)R₂ gives diagonal 1, 3, 58/3. Their product is 58.

Worked example 3

det([[1,2,3],[2,4,6],[0,1,1]])

Dependent rows make the matrix singular.

The second row is twice the first row.
Apply R₂ ← R₂ − 2R₁; the second row becomes zero.
A matrix with a zero row has determinant zero; this row replacement does not change the determinant.

Result: det(A) = 0; no inverse exists.

Check: First-row expansion also gives 1·(4−6) − 2·(2−0) + 3·(2−0) = −2−4+6 = 0.

Common mistakes to catch

Check the conditions before trusting a result.

Dropping cofactor signs

First-row expansion uses +, −, +. Keep the sign separate from the signed minor.

Treating every row operation alike

Replacement preserves the determinant, swaps negate it, and row scaling multiplies it.

Using a square-matrix formula on a rectangle

A determinant requires a square matrix. Sarrus does not extend to 4×4 or larger matrices.