MATRIX WORKSPACE
det(A)
Matrix arithmetic
Core matrix operations
Advanced operations
DIMENSIONS
This operation needs a square matrix.Blank cells count as 0. Fractions such as 1/2 are accepted.
Paste a matrix from a spreadsheet
Matrix A
FREE ADVANCED MATRIX & VECTOR CALCULATOR
Matrix calculator for rank, RREF, inverse, and linear systems.
Build matrices from 2×2 through 10×10, paste values from a spreadsheet, and use arithmetic, row reduction, inverse, determinant, matrix rank with steps, row space checks, eigenvalue, and linear-system tools.3 × 3 A
WHAT THIS MATRIX CALCULATOR CAN SOLVE
Choose the operation that matches your matrix problem.
MATRIX ARITHMETIC
Add, subtract, multiply, scale, and transpose
Use A + B and A − B for equal-size matrices, A × B for compatible dimensions, kA for scalar multiplication, and Aᵀ to swap rows and columns.ROW REDUCTION
REF, RREF, rank, pivots, and row space
Use the row-reduction tools as a matrix reducer calculator. REF applies forward elimination, while RREF continues until each pivot is 1 with zeros above and below it. Rank counts pivot columns, and the non-zero pivot rows help describe the row space. REF is also called row echelon form; in some German study material it appears as Zeilenstufenform.SQUARE-MATRIX TOOLS
Determinant, trace, inverse, cofactor, and adjugate
For square matrices, calculate det(A), tr(A), A⁻¹, a cofactor matrix, and an adjugate. An inverse exists only when the determinant is not zero.LINEAR SYSTEMS
Solve A · x = b with a method you can choose
Solve square coefficient systems with automatic row reduction, Gaussian elimination, Gauss–Jordan RREF, the matrix inverse method, or Cramer’s rule where it applies.ADVANCED RESULTS
Cofactors, adjugates, and 2×2 eigenvalues
Cofactor and adjugate calculations are available for square matrices up to 5×5 to keep the workspace responsive. Eigenvalue mode currently supports 2×2 matrices.FLEXIBLE INPUT
Enter fractions or paste a spreadsheet grid
Enter decimals, scientific notation, or simple fractions such as 1/2. You can also paste a rectangular grid from a spreadsheet; blank cells are treated as zero.RECTANGULAR MATRICES
When rows and columns are different
- Add or subtract matrices with identical dimensions.
- Multiply when columns in A equal rows in B.
- Use scalar multiplication, transpose, REF, RREF, rank, and row-space checks.
SQUARE MATRICES
When rows and columns are the same
- Calculate determinant, trace, inverse, cofactor, and adjugate.
- Solve A · x = b systems with a selected matrix method.
- Use eigenvalue mode for a 2×2 square matrix.
MATRIX RANK CALCULATOR WITH STEPS
Use RREF to find pivot columns, rank, and row space.
Enter the matrix below, choose RREF(A), then compare it with rank(A). The second row is twice the first, so row reduction removes the repeated direction and shows why the matrix has rank 1.LEARN THE METHOD
Read the matching matrix guide.
Use these short guides when you want to understand row operations, pivots, determinants, rank, row space, and linear-system methods before or after checking a result in the calculator.GAUSSIAN ELIMINATIONReduce a system using row operationsLearn the REF and RREF workflow behind matrix reduction and rank checks.Read guide →INVERSE & DETERMINANTCheck when a matrix inverse existsSee how determinant values connect to invertibility, pivots, and solutions.Read guide →LINEAR SYSTEM METHODSCompare Gaussian, RREF, inverse, and Cramer’s ruleChoose a suitable method and interpret rank, pivots, and special cases.Read guide →