How Do I Calculate a 3x3 Determinant?

For a 3×3 matrix [[a,b,c],[d,e,f],[g,h,i]]: det = a(ei-fh) - b(di-fg) + c(dh-eg). This is cofactor expansion along the first row.

The Formula

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

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

Each first-row element multiplies the determinant of its 2×2 minor (the submatrix remaining after deleting that element's row and column). Signs alternate: +, -, +.

Worked Example

    | 2  3  1 |
A = | 4  0  2 |
    | 1  5  3 |
  1. First term (a=2): 2 × (0×3 - 2×5) = 2 × (0 - 10) = 2 × (-10) = -20
  2. Second term (b=3): -3 × (4×3 - 2×1) = -3 × (12 - 2) = -3 × 10 = -30
  3. Third term (c=1): 1 × (4×5 - 0×1) = 1 × (20 - 0) = 1 × 20 = 20
  4. det(A) = -20 + (-30) + 20 = -30

Understanding the Pattern

The cofactor expansion follows a checkerboard sign pattern:

| +  -  + |
| -  +  - |
| +  -  + |

You can expand along any row or column -- the result is always the same. Choosing a row or column with zeros makes computation easier because those terms vanish.

What Does a Zero Determinant Mean?

A determinant of zero means the matrix is singular: it has no inverse, its rows are linearly dependent, and the corresponding system of equations has either no solution or infinitely many solutions.

Use the ML3X matrix calculator to compute determinants for matrices up to 5×5 with full step-by-step cofactor expansion display.

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