Matrix Calculator

For a 2x2 matrix [[a, b], [c, d]], the determinant is (a x d) - (b x c). For a 3x3 matrix, the calculator expands along the first row: a(ei - fh) - b(di - fg) + c(dh - eg). The result is a single number that describes how the matrix scales area or volume, so a determinant of 2 means the transformation doubles area.

For a 2x2 matrix with determinant det, the calculator swaps the diagonal entries, negates the off-diagonal entries, and divides every entry by det. For a 3x3 matrix, it builds the cofactor matrix, transposes it to get the adjugate, then divides each entry by the determinant. The product of a matrix and its inverse is always the identity matrix.

A matrix is invertible when its determinant is not zero. A zero determinant means the rows are linearly dependent, so the matrix collapses space into a lower dimension and no inverse can restore the original. This calculator supports 2x2 and 3x3 square matrices only; if the determinant is zero, it returns an error instead of a result.

The transpose flips a matrix over its main diagonal, turning each row i into column i. A 2x3 matrix becomes a 3x2 matrix after transposing. Common uses include switching between row and column vectors, building symmetric matrices (A x A^T is always symmetric), and solving least-squares regression problems in statistics.

The trace is the sum of the entries on the main diagonal, top-left to bottom-right. For [[1, 2], [3, 4]] the trace is 1 + 4 = 5. It equals the sum of the matrix's eigenvalues and stays unchanged under similarity transformations, including a change of basis. Rotating the matrix entries without applying the corresponding inverse transformation does not preserve trace.