How to Calculate a Matrix Determinant

The determinant is a scalar value that can be computed from a square matrix. It appears in linear algebra when solving systems of equations, finding matrix inverses, and understanding linear transformations. If the determinant is zero, the matrix is "singular" and has no inverse.

2×2 Matrix Determinant

For matrix:

|a  b|
|c  d|

det = ad − bc

Example: det([[3, 1], [5, 2]]) = (3×2) − (1×5) = 6 − 5 = 1

3×3 Matrix Determinant (Cofactor Expansion)

For matrix:

|a  b  c|
|d  e  f|
|g  h  i|

det = a(ei − fh) − b(di − fg) + c(dh − eg)

Example:

|2  1  3|
|0  4  1|
|5  2  6|

det = 2(4×6 − 1×2) − 1(0×6 − 1×5) + 3(0×2 − 4×5) = 2(24 − 2) − 1(0 − 5) + 3(0 − 20) = 2(22) − 1(−5) + 3(−20) = 44 + 5 − 60 = −11

Properties of Determinants

  • det(AB) = det(A) × det(B)
  • det(Aᵀ) = det(A)
  • Swapping two rows changes the sign of the determinant
  • If two rows are identical, det = 0
  • Multiplying a row by k multiplies the determinant by k

Use our matrix determinant calculator for any square matrix.