Matrix Calculator

The Matrix Calculator performs matrix operations and writes down each one. Addition, scalar multiplication, and matrix products with the dimension check. Transpose. Determinant by cofactor expansion or row reduction, with the expansion row chosen and stated. Inverse by the adjugate method or by augmenting with the identity and row reducing. Rank and reduced row echelon form. Solving Ax = b. Eigenvalues from the characteristic polynomial and eigenvectors from the null space of A minus lambda I. Diagonalization when it exists. Cramer's rule, LU decomposition, and the matrix form of a linear transformation, including rotation and reflection matrices, are also supported, with the determinants or factors shown.

Enter matrices in bracket notation: [[2,1],[5,3]] for a 2 by 2, rows separated by commas inside the outer brackets. Or photograph the printed matrix. State the operation: "find the inverse", "determinant", "RREF", "eigenvalues and eigenvectors", "solve Ax = b with b = [7, -1]". If the course requires a specific method, name it. Set Answer form to exact for fractions and radicals or decimal for rounded entries. For a linear transformation question, state the transformation in words or give where the basis vectors map, and the solver builds the matrix. For a system, enter the coefficient matrix and the constant vector separately, or just paste the equations and ask for the matrix method.

Output: the resulting matrix, value, or vector first. Then every intermediate matrix with the operation that produced it, written as R2 -> R2 - 3R1 or "expand along row 1". Eigenvalue problems show the characteristic polynomial and its roots before the eigenvectors. A final check multiplies the inverse by the original, or substitutes the solution back. The working is shown so you can check it. For a small system where matrices are optional, the System of Equations Solver also does substitution and elimination. A follow-up can ask for the determinant by a different method as a check, or for the inverse verified by multiplication. Fraction entries are kept exact unless you ask for decimals, so the working matches what a textbook key prints.