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.

Det [m] gives the determinant of a square matrix m. Minors [m] is the matrix whose ^(th) element gives the determinant of the submatrix obtained by deleting the ^(th) row and the ^(th) column of m. The ^(th) cofactor of m is times the ^(th) element of the matrix of minors.

Minors 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.

This generates a 3×3 matrix, whose ^(th) entry is .

Here is the determinant of .

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.

The rank of a matrix is the number of linearly independent rows or columns.

This finds the rank of a matrix.

Here is a 2×2 matrix.

This gives the third matrix power of .

It is equivalent to multiplying three copies of the matrix.

Here is the millionth matrix power.

The matrix exponential of a matrix m is , where indicates a matrix power.

This gives the matrix exponential of .

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