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## Matrices & Linear Algebra

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

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Matrix Arithmetic

Add, subtract and multiply vectors and matrices.

multiply matrices

matrix vector product

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Trace

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

compute the trace of a matrix

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Determinant

Calculate the determinant of a square matrix.

compute the determinant of a matrix

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Inverse

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

compute the inverse of a matrix

find a pseudoinverse

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Row Reduction

Reduce a matrix to its reduced row echelon form.

row reduce a matrix

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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

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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

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Diagonalization

Find the diagonalization of a square matrix.

diagonalize a matrix

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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

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Vector Spaces

Compute properties of linear vector spaces.

compute the row space of a matrix

compute the column space of a matrix

compute the null space of a matrix

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Linear Independence

Check vectors for both linear dependence and linear independence.

determine whether a set of vectors is linearly independent

specify complex vectors

specify vectors with one or more symbolic components

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