Examples for

# Matrices

A matrix is a two-dimensional array of values that is often used to represent a linear transformation or a system of equations. Matrices have many interesting properties and are the core mathematical concept found in linear algebra and are also used in most scientific fields. Matrix algebra, arithmetic and transformations are just a few of the many matrix operations at which Wolfram|Alpha excels.

Explore various properties of a given matrix.

#### Calculate properties of a matrix:

Calculate the trace or the sum of terms on the main diagonal of a matrix.

#### Compute the trace of a matrix:

Invert a square invertible matrix or find the pseudoinverse of a non-square matrix.

#### Compute the inverse of a matrix:

#### Find a pseudoinverse:

Perform various operations, such as conjugate transposition, on matrices.

#### Compute the transpose of a matrix:

#### Compute the rank of a matrix:

#### Compute the nullity of a matrix:

#### Compute the adjugate of a matrix:

#### Compute the minors of a matrix:

Find matrix representations for geometric transformations.

#### Compute a 2 x 2 rotation matrix:

#### Compute a 3 x 3 reflection matrix:

Add, subtract and multiply vectors and matrices.

#### Add matrices:

#### Multiply matrices:

#### Matrix vector product:

Calculate the determinant of a square matrix.

#### Compute the determinant of a matrix:

Reduce a matrix to its reduced row echelon form.

#### Row reduce a matrix:

Explore diagonalizations, such as unitary and orthogonal diagonalizations, of a square matrix.

#### Diagonalize a matrix:

#### Compute the orthogonal diagonalization of a matrix:

#### Calculate the unitary diagonalization of a matrix:

Find information on many different kinds of matrices.

#### Determine whether a matrix has a specified property:

#### Get information about a type of matrix:

#### Specify a size:

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Find a specific minor of a square matrix.

#### Calculate a minor of a matrix:

Calculate the eigensystem of a given matrix.

#### Compute the eigenvalues of a matrix:

#### Compute the eigenvectors of a matrix:

#### Compute the characteristic polynomial of a matrix:

Transform a matrix into a specified decomposition.