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:

{{6, -7}, {0, 3}}
{{1, -5, 8}, {1, -2, 1}, {2, -1, -5}}
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Matrix Arithmetic

Add, subtract and multiply vectors and matrices.

Add matrices:

{{1, 2}, {3, 4}} + {{2, -1}, {-1, 2}}

Multiply matrices:

{{2, -1}, {1, 3}} . {{1, 2}, {3, 4}}

Matrix vector product:

{{2, -1, 1}, {0, -2, 1}, {1, -2, 0}} . {x, y, z}
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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:

tr {{9, -6, 7}, {-9, 4, 0}, {-8, -6, 4}}
tr {{a, b}, {c, d}}
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Determinant

Calculate the determinant of a square matrix.

Compute the determinant of a matrix:

determinant of {{3, 4}, {2, 1}}
det({{9, 3, 5}, {-6, -9, 7}, {-1, -8, 1}})
det {{a, b, c}, {d, e, f}, {g, h, j}}
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Minors

Find a specific minor of a square matrix.

Calculate a minor of a matrix:

(1,2)-minor of {{1,2,3},{3,4,5},{5,4,3}}
M-23 of {{2, a, 5, b}, {7, c+d, 9, 11}, {13, 17, 19, 23}, {a-1, -2, b+c, 4}}
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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:

inv {{10, -9, -12}, {7, -12, 11}, {-10, 10, 3}}
inverse {{a, b}, {c, d}}
{{2, 3}, {4, 5}}^(-1)

Find a pseudoinverse:

inverse {{1, -4, 3}, {2, -5, 8}}
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Row Reduction

Reduce a matrix to its reduced row echelon form.

Row reduce a matrix:

row reduce {{2, 1, 0, -3}, {3, -1, 0, 1}, {1, 4, -2, -5}}
row reduction calculator
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Eigenvalues & Eigenvectors

Calculate the eigensystem of a given matrix.

Compute the eigenvalues of a matrix:

eigenvalues {{4, 1}, {2, -1}}

Compute the eigenvectors of a matrix:

eigenvectors {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}}

Compute the characteristic polynomial of a matrix:

characteristic polynomial {{4, 1}, {2, -1}}
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Other Matrix Operations

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

Compute the transpose of a matrix:

transpose {{-3, 2}, {5, 1}}

Compute the rank of a matrix:

rank {{6, -11, 13}, {4, -1, 3}, {3, 4, -2}}

Compute the nullity of a matrix:

nullity {{6, -11, 13}, {4, -1, 3}, {3, 4, -2}}

Compute the adjugate of a matrix:

adjugate {{8, 7, 7}, {6, 9, 2}, {-6, 9, -2}}

Compute the minors of a matrix:

minors {{1, 2, 3}, {3, 2, 1}, {2, 1, 3}}
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Matrix Decompositions

Transform a matrix into a specified decomposition.

Diagonalize a matrix:

diagonalize {{1, 2}, {3, 4}}

Compute the LU decomposition of a square matrix:

LU decomposition of {{7, 3, -11}, {-6, 7, 10}, {-11, 2, -2}}

Compute a singular value decomposition:

SVD {{1, 0, -1}, {-2, 1, 4}}
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Geometric Transformations

Find matrix representations for geometric transformations.

Compute a 2 x 2 rotation matrix:

rotate 30 degrees

Compute a 3 x 3 reflection matrix:

reflect across x + y + z = 1
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Types of Matrices

Find information on many different kinds of matrices.

Determine whether a matrix has a specified property:

Is {{3, -3}, {-3, 5}} positive definite?

Get information about a type of matrix:

Hilbert matrices
Hankel matrices

Specify a size:

5x5 Hilbert matrix
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