Find the row reduction of a real machine-number matrix:

Row reduce a complex machine-precision matrix:

Row reduce an arbitrary-precision matrix:

Row reduce an exact matrix:

Row reduction of a symbolic matrix:

RowReduce assumes all symbols to be independent:

Row reduction of rectangular matrices:

The row reduction of a large numerical matrix is computed efficiently:

Row reduction of a sparse matrix:

Row reduction of a structured matrix:

RowReduce does not alter an identity matrix:

Row reduction of a Hilbert matrix:

Compute the row reduction of a matrix of univariate polynomials of degree :

m is a 3×3 matrix of integers between 0 and 4:

Row reduction of m in ordinary arithmetic:

Row reduction of m in arithmetic modulo 5:

m is an ill-conditioned matrix:

In exact arithmetic, m is clearly non-degenerate:

With machine arithmetic, the default is to consider elements that are too small as zero:

With zero tolerance, even small terms may be taken into account:

With an augmented matrix, you can see how possible solution components are amplified:

By default, symbolic expressions are considered nonzero:

In this case, RowReduce is missing the special case that produces a singular matrix:

Write a function that tests if an expression might be zero:

Pass test[k] as the value of ZeroTest to get a more symbolic reduction:

This catches the special case at , but gives a non-reduced matrix for other values of :

The following three vectors are not linearly independent:

The row-reduced form has a zero row:

The following three vectors are linearly independent:

Therefore the row-reduced form is the identity matrix:

Determine if the following vectors are linearly independent or not:

As does not reduce to an identity matrix, they are not linearly independent:

Find the dimension of the column space of the following matrix:

Since the row-reduced form is an identity matrix, the dimension of the column space equals the number of columns:

Find the dimension of the subspace spanned by the following vectors:

Since the row-reduced form has three nonzero rows, that is the dimension of the subspace:

Determine if the following system of equations has a unique solution:

Rewrite the system in matrix form:

The coefficient matrix reduces to the identity matrix, so the system has a unique solution:

Verify the result using Solve:

Solve the system , with a matrix and a vector, using row reduction:

Form an augmented matrix :

Do row reduction on the augmented matrix:

The last column is the solution of :

Do it for another right‐hand side:

There is no solution to since there is a leading 1 in the last column; confirm using Solve:

Find all solutions of the following system of equations:

First, write the coefficient matrix , variable vector and constant vector :

Verify the rewrite:

Reduce the augmented matrix to find one solution:

Extract the last column, which forms one solution:

Since there is a zero row, the null space is nonempty. Row reduce the augmented matrix :

The last half of the last row is an element of the null space:

The general solution is the sum of and any multiple of :

Determine if the following matrix has an inverse:

It does not reduce to an identity matrix, so it is not invertible:

Verify the result using Inverse:

Determine if the following matrix has a nonzero determinant:

Since it reduces to an identity matrix, its determinant must be nonzero:

Confirm the result using Det:

is an eigenvalue of if does not reduce to an identity matrix. A matrix is deficient if it has an eigenvalue whose multiplicity is greater than the number of zero rows in the row-reduced form. Show that is an eigenvalue for the following matrix :

Confirm the result using Eigenvalues:

The matrix is deficient because 2 appears twice, but the row-reduced form only has one zero row:

Confirm the result with Eigensystem, which indicates deficiency by padding the eigenvector list with zeros:

Find the inverse of the following matrix using row reduction:

Form the augmented matrix :

Row-reduce the augmented matrix:

The first three columns of are the identity matrix:

The last three columns are the inverse of :

Verify the result using Inverse:

Estimate the fraction of random 10×10 0–1 matrices that are invertible: