Linear Algebra Intermediate

Determinant Calculator

Compute det(A) for 2×2 and 3×3 matrices with full cofactor expansion steps. See every minor, every cofactor, and the Sarrus rule visualized on the canvas.

Subject: Linear Algebra
Input: 2×2 or 3×3 matrix
Output: det(A) + cofactor steps
Live
Matrix Input
Result
Enter matrix values and click Calculate

Step-by-Step Work

What the Determinant Tells You
det(A) = ad − bc  (2×2)

The determinant encodes how much a matrix scales area (2D) or volume (3D). If det(A) = 0 the matrix is singular — its columns are linearly dependent and the transformation collapses space to a lower dimension.

For a 3×3 matrix, cofactor expansion picks any row or column and sums products of each entry with its signed minor. Expanding along the first row is most common.

Exam tip — swapping two rows changes the sign of the determinant. Adding a multiple of one row to another leaves it unchanged.
Key Properties
  • det(AB) = det(A) · det(B)
  • det(Aáµ€) = det(A)
  • det(kA) = kⁿ · det(A) for n×n
  • det(A⁻¹) = 1 / det(A)
You Might Also Need
Matrix Inverse (A⁻¹)
Find the inverse using adjugate or row reduction
Open tool →
Matrix Rank Calculator
Determine rank via row reduction
Open tool →
RREF Calculator
Full row reduction with every step shown
Open tool →

Still stuck? Let's work through it together.

"Bring this exact matrix to a session — I'll help you see why cofactor expansion works and how determinants connect to geometry."

Book a Free Consultation →