Calculate the reduced row echelon form (RREF) of matrices. Perfect for linear algebra, matrix theory, and systems of equations.
Enter matrix elements separated by commas, rows separated by new lines
Comprehensive matrix calculations with detailed explanations
Calculate reduced row echelon form
Calculate matrix rank and nullity
Solve linear systems of equations
Test vector linear independence
Understanding matrix operations and linear algebra concepts
Enter your matrix elements or system of equations.
Use elementary row operations to reduce the matrix.
Receive RREF matrix, rank, nullity, and solution analysis.
Leading coefficients = 1, zeros below and above
rank(A) + nullity(A) = number of columns
Unique, infinite, or no solution
Elementary row operations
Common matrix calculations and their RREF results
Result: RREF with pivot columns
Result: Already in RREF form
Result: 3D transformations
Result: Current/voltage values
Common questions about reduced echelon form calculations
RREF is a canonical form of a matrix where: leading coefficients are 1, leading coefficients are the only non-zero entries in their columns, and rows with all zeros are at the bottom.
Row Echelon Form (REF) has leading coefficients of 1 and zeros below them. RREF additionally has zeros above the leading coefficients, making it unique for each matrix.
The rank of a matrix is the number of non-zero rows in its RREF form, or equivalently, the number of pivot columns. It represents the dimension of the column space.
The nullity of a matrix is the dimension of its null space (kernel). It's calculated as nullity = number of columns - rank, following the rank-nullity theorem.
Form an augmented matrix [A|b], reduce it to RREF, then read the solutions directly from the RREF form. The system has a unique solution if rank(A) = rank([A|b]) = number of variables.
Elementary row operations are: 1) Swap two rows, 2) Multiply a row by a non-zero scalar, 3) Add a multiple of one row to another row. These operations preserve the solution set.
Form a matrix with the vectors as columns, reduce to RREF. The vectors are linearly independent if and only if every column is a pivot column (rank = number of vectors).
A system has infinite solutions when rank(A) = rank([A|b]) < number of variables. The free variables can take any value, while basic variables are expressed in terms of free variables.
Pivot columns are those containing leading coefficients (the first non-zero entry in each row) in the RREF form. They correspond to basic variables in the solution.
RREF is used in computer graphics (transformations), electrical engineering (circuit analysis), economics (input-output models), machine learning (feature selection), and cryptography (linear codes).
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