Matrix Determinant Calculator
Calculate the determinant of a 2x2 or 3x3 matrix with step-by-step work.
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Request a ToolHow to Use the Matrix Determinant Calculator
Select a matrix size, fill in the values, and the determinant appears instantly with step-by-step work.
- Choose matrix size. Select 2x2 or 3x3.
- Enter matrix values. Fill in each cell. The grid auto-generates based on size. Default is the identity matrix.
- Read the result. The determinant is shown along with whether the matrix is singular (det = 0) or invertible (det != 0). Detailed steps are shown below.
About Matrix Determinants
The determinant is a scalar value computed from a square matrix that encodes important properties. For a 2x2 matrix [[a,b],[c,d]], the determinant is ad - bc. For 3x3 matrices, cofactor expansion along the first row is used.
A determinant of zero means the matrix is singular (not invertible), and the system of equations it represents has no unique solution. A non-zero determinant means the matrix is invertible and the system has a unique solution. Determinants are used in Cramer's rule, eigenvalue calculations, and area/volume computations.
Frequently Asked Questions
What does a zero determinant mean?
A zero determinant means the matrix is singular and has no inverse. In terms of linear equations, the system either has no solution or infinitely many solutions.
How do you calculate a 2x2 determinant?
For a 2x2 matrix [[a,b],[c,d]], the determinant is ad - bc. Multiply the diagonals and subtract: top-left times bottom-right minus top-right times bottom-left.
How do you calculate a 3x3 determinant?
Use cofactor expansion along the first row. Multiply each element by its cofactor (the determinant of the 2x2 submatrix formed by removing that row and column), alternating signs: +a(ei-fh) - b(di-fg) + c(dh-eg).