Matrix Calculator
Compute determinant, inverse, transpose, trace, and add, subtract, or multiply 2x2 and 3x3 matrices.
Matrix operations
Result
-2
Frequently Asked Questions about the Matrix Calculator
How do I calculate the determinant of a matrix?
For a 2x2 matrix [[a, b], [c, d]], the determinant is ad minus bc. For a 3x3 matrix, expand along the first row: a(ei - fh) - b(di - fg) + c(dh - eg). The determinant is a single number that summarizes how the matrix scales space.
How do I find the inverse of a matrix?
The inverse equals 1 divided by the determinant, multiplied by the adjugate (the transpose of the cofactor matrix). The calculator returns the inverse only when the determinant is non-zero. The product of a matrix and its inverse is the identity matrix.
When is a matrix invertible?
A square matrix is invertible exactly when its determinant is not zero. If the determinant is zero, the matrix is singular: its rows are linearly dependent and no inverse exists. The calculator shows an error message in that case.
What is a transpose and when do I use it?
The transpose flips a matrix over its diagonal, so row i becomes column i. It is used to switch between row and column vectors, to build symmetric matrices via A times its transpose, and to define inner products and least-squares fits in statistics and machine learning.
What is the trace of a matrix?
The trace is the sum of the entries on the main diagonal of a square matrix. It equals the sum of the eigenvalues and is invariant under change of basis, which makes it useful in physics, statistics, and matrix calculus.