Matrix Determinant Calculator
Calculate the determinant of 2x2 and 3x3 matrices with step-by-step working.
Uses cofactor expansion and applies to linear algebra and eigenvalue problems.
The determinant is a single number calculated from a square matrix. It tells you whether the matrix is invertible and describes the scaling factor of the linear transformation the matrix represents. In geometry, the absolute value of the determinant gives the factor by which areas (2D) or volumes (3D) are scaled.
2x2 Matrix:
For matrix [[a, b], [c, d]]:
det = ad - bc
Practical Example (2x2): For the matrix [[3, 7], [1, 5]]: det = (3)(5) - (7)(1) = 15 - 7 = 8 Since 8 is not 0, this matrix is invertible.
3x3 Matrix (cofactor expansion along first row):
For matrix [[a, b, c], [d, e, f], [g, h, i]]:
det = a(ei - fh) - b(di - fg) + c(dh - eg)
This formula works by breaking the 3x3 problem into three 2x2 determinants (called minors), each multiplied by the corresponding element from the first row, with alternating signs (+, -, +).
When to use this calculator: Determinants appear in many areas of mathematics, science, and engineering. They are used to solve systems of linear equations (Cramer’s Rule), find inverse matrices, compute eigenvalues, and test whether a set of vectors is linearly independent.
Key properties:
- If det = 0, the matrix is singular (not invertible, no unique solution to the system)
- If det is not 0, the matrix is invertible and the system has a unique solution
- det(AB) = det(A) x det(B)
- Swapping two rows negates the determinant
- Multiplying a row by a constant k multiplies the determinant by k
- Adding a multiple of one row to another row does not change the determinant
Common Mistakes:
- Mixing up the sign pattern in cofactor expansion: remember it alternates: +, -, +
- Confusing rows and columns when reading matrix entries
- Forgetting that a 0 determinant means the system of equations has either no solution or infinitely many solutions, not a single answer