Find the inverse of a machine-precision matrix:

Invert a complex matrix:

Inverse of an exact matrix:

Inverse of an arbitrary-precision matrix:

Inverse of a symbolic matrix:

Verifying a symbolic inverse may require simplification:

The inversion of large machine-precision matrices is efficient:

TraditionalForm formatting:

The inverse of a sparse matrix is returned as a normal matrix:

Format the result:

When possible, the inverse of a structured matrix is returned as another structured matrix:

This is not always possible:

IdentityMatrix is its own inverse:

Inverse of HilbertMatrix:

Visualize the inverses for several matrix sizes:

Invert a matrix using arithmetic modulo five:

The inverse using normal arithmetic:

Visualize the two results:

The automatic zero test cannot detect that the following matrix is nonsingular:

The problem is that machine-precision underflows for the bottom right entry:

The entry is very small but nonzero:

Use a zero test employing arbitrary-precision arithmetic to invert the matrix:

Solve the system of equations , , . First, form the coefficient matrix and constant vector :

The solution is given by :

Substitute the solution into the original system of equations to verify the solution:

Solve the matrix equation :

Multiplying both sides of the equation on the left by shows :

Confirm the result using LinearSolve:

For numerical and especially sparse systems, LinearSolve can be considerably faster:

Solve the matrix equation :

Multiplying both sides of the equation on the left by shows :

Substitute the solution into the equation for verification:

Solve the system of ODEs , , . First, construct the coefficient matrix for the right-hand side:

Find the eigenvalues and eigenvectors:

Construct a diagonal matrix whose entries are the exponential of :

Construct the matrix whose columns are the corresponding eigenvectors:

The general solution is , for three arbitrary starting values:

Verify the solution using DSolveValue:

Express a general vector in as a linear combination of the vectors , and . First, verify the vectors are linearly independent by checking that their null space is empty:

Form the matrix whose columns are the basis vectors:

The coefficients of a general vector are given by :

Verify that does indeed equal the linear combination :

Find the change-of-basis matrix that transforms coordinates with respect to the basis to coordinates with respect to the basis :

The matrix whose columns are the transforms from -coordinates to standard coordinates:

The matrix whose columns are the transforms from -coordinates to standard coordinates:

Its inverse converts from standard coordinates back to -coordinates:

Therefore, converts from -coordinates to -coordinates:

Express the linear operator whose representation in the standard is given by in the basis , ,

The matrix whose columns are the transforms from -coordinates to standard coordinates:

Its inverse converts from standard coordinates to -coordinates:

Therefore the representation of in -coordinates is:

Inverse can be used to diagonalize a matrix as . Compute 's eigenvalues and eigenvectors:

Construct a diagonal matrix from the eigenvalues and a matrix whose columns are the eigenvectors:

Confirm the identity :

Any function of the matrix can now be computed as . For example, MatrixPower:

Similarly, MatrixExp becomes trivial, requiring only exponentiating the diagonal elements of :

Inverse of a transformation matrix gives the matrix for the reverse operation. For example, consider a translation by :

The inverse of its transformation matrix gives a translation by the opposite motion:

Consider a general affine transformation:

Construct the inverse transformation:

Verify that the two transformations really do undo each other:

For a mapping , the Jacobian of the inverse mapping is given by . Consider the mapping from Cartesian to spherical coordinates:

Compute the Jacobian at the point :

The inverse mapping is the transformation from spherical back to Cartesian coordinates:

Verify the inverse function:

Compute the Jacobian of the inverse mapping at the coordinates corresponding to :

Confirm the identity :