# 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] 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:
 In:= Out= 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:
 In:= Out= This generates a 3×3 matrix, whose entry is a[i,j]:
 In:= Out= Here is the determinant of m:
 In:= Out= 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:
 In:= Out= The rank of a matrix is the number of linearly independent rows or columns.

This finds the rank of a matrix:
 In:= Out= MatrixPower[m,n] n matrix power MatrixExp[m] matrix exponential

Powers and exponentials of matrices.

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

This gives the matrix exponential of m:
 In:= Out= Here is an approximation to the exponential of m, based on a power series approximation:
 In:= Out= 