# Basic Matrix Operations

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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Powers and exponentials of matrices.

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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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