Calculate the determinant |A| and trace for 2x2 and 3x3 matrices instantly with step-by-step expansion formulas.
Calculate the determinant det(A) and trace for 2x2 and 3x3 matrices using cofactor expansion formulas.
Understanding Matrix Determinants
The determinant of a square matrix is a single scalar value that summarizes key geometrical and algebraic properties of the linear transformation represented by the matrix.
Applications of Determinants
Matrix Invertibility: A matrix has an inverse if and only if its determinant is non-zero (det(A) ≠ 0).
Solving Linear Systems: Used in Cramer’s Rule to solve system of linear equations.
Area and Volume Scaling: Represents the scaling factor of area (in 2D) or volume (in 3D) transformed by the matrix.
How to use
Select matrix dimensions (2x2 or 3x3).
Enter matrix element values.
View the calculated determinant det(A) and matrix trace tr(A).
Formula
2x2 Determinantdet(A) = a11·a22 - a12·a21Cross-multiplication of 2x2 diagonal elements.
3x3 Determinantdet(A) = a11(a22·a33 - a23·a32) - a12(a21·a33 - a23·a31) + a13(a21·a32 - a22·a31)Cofactor expansion across the first row.
Worked examples
2x2 Determinant Calculation
For A = [[4, 3], [2, 5]], det(A) = (4×5) - (3×2) = 20 - 6 = 14.
Frequently asked questions
What is a matrix determinant?
A determinant is a scalar property of a square matrix used to solve systems of linear equations, invert matrices, and calculate volumes in geometry.
What does a zero determinant mean?
If det(A) = 0, the matrix is singular and non-invertible (it does not have a matrix inverse).