# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
WOLFRAM LANGUAGE TUTORIAL

# Basic Matrix Operations

 Transpose[m] transpose ConjugateTranspose[m] conjugate transpose (Hermitian conjugate) Inverse[m] matrix inverse Det[m] determinant Minors[m] matrix of minors Minors[m,k] k minors Tr[m] trace MatrixRank[m] rank of matrix

Some basic matrix operations.

Transposing a matrix interchanges the rows and columns in the matrix. If you transpose an m×n matrix, you get an n×m matrix as the result.

Transposing a 2×3 matrix gives a 3×2 result.
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Det[m] gives the determinant of a square matrix m. Minors[m] is the matrix whose element gives the determinant of the submatrix obtained by deleting the row and the column of m. The cofactor of m is times the element of the matrix of minors.

Minors[m,k] gives the determinants of the k×k submatrices obtained by picking each possible set of k rows and k columns from m. Note that you can apply Minors to rectangular, as well as square, matrices.

Here is the determinant of a simple 2×2 matrix.
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This generates a 3×3 matrix, whose entry is .
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Here is the determinant of .
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The trace or spur of a matrix Tr[m] is the sum of the terms on the leading diagonal.

This finds the trace of a simple 2×2 matrix.
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The rank of a matrix is the number of linearly independent rows or columns.

This finds the rank of a matrix.
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 MatrixPower[m,n] n matrix power MatrixExp[m] matrix exponential

Powers and exponentials of matrices.

Here is a 2×2 matrix.
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This gives the third matrix power of .
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It is equivalent to multiplying three copies of the matrix.
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Here is the millionth matrix power.
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The matrix exponential of a matrix m is , where indicates a matrix power.

This gives the matrix exponential of .
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Here is an approximation to the exponential of , based on a power series approximation.
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