Power Calculators

|A| Determinant Calculator (Cofactor Expansion)

det(A) with the working — and whether the matrix is invertible.

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Formula at a glance

Determinant Calculator: 2 × 2 — det = ad − bc. Example: 3 × 3 matrix → det(A): 49
How the Determinant Calculator works: 2 × 2 — det = ad − bc

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📘 About the Determinant Calculator

The determinant is a single number computed from a square matrix. It tells you whether the matrix is invertible (det ≠ 0), how much it scales areas or volumes, and whether it flips orientation (det < 0). Determinants solve equations by Cramer’s rule and give eigenvalues through the characteristic polynomial.

Enter a square matrix. For 2 × 2 the calculator shows ad − bc; for 3 × 3 and 4 × 4 it expands along the first row with each cofactor; larger matrices are reduced to triangular form. The answer is exact.

🧮 Formula

2 × 2
det = ad − bc
Cofactor expansion
det(A) = Σⱼ a₁ⱼ·C₁ⱼ, C₁ⱼ = (−1)^(1+j)·M₁ⱼ
Properties
det(AB) = det(A)·det(B), det(Aᵀ) = det(A)

✏️ Worked examples

3 × 3 matrix

You enter
Matrix A2 -3 1; 2 0 -1; 1 4 5
You get
det(A)49
Invertible?Yes — det ≠ 0
A[2, −3, 1; 2, 0, −1; 1, 4, 5]
Geometric meaningVolume scale factor |det| = 49

det = 49.

2 × 2 matrix

You enter
Matrix A1 2; 3 4
You get
det(A)−2
Invertible?Yes — det ≠ 0
A[1, 2; 3, 4]
Geometric meaningArea scale factor |det| = 2

det = −2.

🧭 How to use this calculator

What you enter

  • Matrix ARows separated by ; or new lines, entries by spaces or commas. Fractions like 1/2 are fine.

What you get

  • det(A)
  • Invertible?
  • A
  • Geometric meaning

❓ Frequently asked questions

What does a determinant of 0 mean?

The matrix is singular: it has no inverse, its rows are dependent, and it squashes space into a lower dimension.

How do you find a 3 × 3 determinant?

Expand along a row: multiply each entry by its cofactor (the 2 × 2 determinant left after deleting its row and column, with a ± sign) and add.

What happens if I swap two rows?

The determinant changes sign. Multiplying a row by k multiplies the determinant by k.

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