Tools Matrix

Matrix Determinant and Inverse Calculator

Online calculator to quickly find a variety of matrix properties, including inverses, determinants and the cofactor matrix in one easy to use calculator. Use any square matrix from 2x2, 3x3 all the way up to 10x10. Results will automatically update as you type. For more complex linear algebra calculations you can use our matrix calculator.

Fractions are supported. They are entered as a/b.

(
)

Determinant

1

Trace

3

Inverse

(
1 0 0 0 1 0 0 0 1
)
Matrix is not invertible

Transpose

(
1 0 0 0 1 0 0 0 1
)

Cofactor

(
1 0 0 0 1 0 0 0 1
)

Understanding Matrix Properties

Determinant

The determinant of a matrix is a very important concept in linear algebra. The determinant of a matrix, A, is usually written as either |A| or det(A). It is a scalar value for any matrix. If the matrix represents a linear transformation it gives the corresponding scale factor and any orientation change.

There are many methods to calculate the determinant of a matrix, including Gaussian elimination and by decomposition methods such as LU, QR or Cholesky. This website uses the Laplace expansion which calculates the determinant recursively from the determinant of its minor matrices.

Inverse

The inverse of a matrix is another matrix of the same size that it multiplies with to give the identity matrix. If a matrix has a determinant of 0 it is not invertible.

Trace

The trace is the sum of all the values on the main diagonal of a matrix. It is equal to the sum of the eigenvalues of a vector.

Cofactor

The cofactor matrix, C, is calculated by setting Cij=(-1)i+j|Mij|, where Mij is the minor matrix. The minor matrix is formed by removing the row and column.

The cofactor matrix is the transpose of the adjugate matrix.