3x3 Determinant Calculator

For matrix [[a,b,c],[d,e,f],[g,h,i]]:

(1) Multiply a by the 2x2 minor (ei-fh).

(2) subtract b times the minor (di-fg).

(3) add c times the minor (dh-eg). Example: [[1,2,3],[0,4,5],[1,0,6]] gives 1(24-0) - 2(0-5) + 3(0-4) = 24 + 10 - 12 = 22.

A determinant of zero means the matrix is singular (not invertible). The rows are linearly dependent, meaning one row is a combination of the others. For a system of equations Ax = b, det(A) = 0 means the system has either no solution or infinitely many solutions, never a unique one.

No. Other methods include: row reduction to upper triangular form (then multiply the diagonal), the Sarrus rule (a shortcut specific to 3x3), and the Leibniz formula. For hand calculation, cofactor expansion is the most common. For larger matrices, row reduction is more efficient.

Yes. You can expand along any row or column and get the same result. Choosing a row or column with zeros simplifies the calculation because those terms drop out. For [[1,2,3],[0,4,5],[1,0,6]], expanding along column 1 is easier because it contains a zero.

Determinants are used to: check if a matrix is invertible (det not zero), solve systems of linear equations (Cramer's rule), compute cross products in 3D vectors, find eigenvalues, calculate area (2x2) and volume (3x3) scaling, and determine orientation of geometric transformations.

Yes, the 3x3 Determinant Calculator is fully embeddable. Tap "Embed" above to configure appearance and copy the code. It is free to use, works on any platform (HTML, WordPress, CMS), and adjusts to any screen size automatically. Visit calculory.com/services/embed-calculators for the complete guide.