Matrix Determinant & Inverse (2x2 and 3x3)
Calculate determinants of 2x2 and 3x3 matrices with step-by-step cofactor expansion, matrix trace, invertibility check, and full algebraic formulas.
Calculated Results
Frequently Asked Questions
What does a matrix determinant represent geometrically?
Geometrically, the determinant of a 2x2 matrix represents the scaling factor of the area transformed by the linear transformation. For a 3x3 matrix, the absolute value of the determinant represents the volume of the parallelepiped formed by its column vectors.
How do you calculate the determinant of a 2x2 matrix?
For a 2x2 matrix [[a, b], [c, d]], the determinant is simply: det(A) = (a × d) - (b × c).
What does it mean if the determinant of a matrix is zero?
If det(A) = 0, the matrix is 'singular'. This means the transformation collapses dimensions (e.g. collapses 3D space into a 2D plane or line), the matrix is not invertible, and a system of linear equations Ax = b either has no solutions or infinitely many solutions.
What is the matrix trace tr(A)?
The trace of a square matrix is the sum of all elements along the main top-left to bottom-right diagonal. The trace is invariant under change of basis and equals the sum of all eigenvalues of the matrix.