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Basic Matrix Operations

Transpose[m]transpose m
ConjugateTranspose[m]conjugate transpose m (Hermitian conjugate)
Inverse[m]matrix inverse
Det[m]determinant
Minors[m]matrix of minors
Minors[m,k]kth 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 (i, j)th element gives the determinant of the submatrix obtained by deleting the (n-i+1)th row and the (n-j+1)th column of m. The (i, j)th cofactor of m is (-1)i+j times the (n-i+1, n-j+1)th 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 i, jth entry is a[i, j].
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Here is the determinant of m.
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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]nth 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 m.
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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 mk/k!, where mk indicates a matrix power.
This gives the matrix exponential of m.
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Here is an approximation to the exponential of m, based on a power series approximation.
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