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.

### Matrix Properties

Explore various properties of a given matrix.

#### Calculate properties of a matrix:

### Trace

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

#### Compute the trace of a matrix:

### Inverse

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

#### Compute the inverse of a matrix:

#### Find a pseudoinverse:

### Other Matrix Operations

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:

### Matrix Arithmetic

Add, subtract and multiply vectors and matrices.

#### Add matrices:

#### Multiply matrices:

#### Matrix vector product:

### Determinant

Calculate the determinant of a square matrix.

#### Compute the determinant of a matrix:

### Row Reduction

Reduce a matrix to its reduced row echelon form.

#### Row reduce a matrix:

### Diagonalization

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:

### Types of Matrices

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:

### GO FURTHER

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### Minors

Find a specific minor of a square matrix.

#### Calculate a minor of a matrix:

### Eigenvalues & Eigenvectors

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.