Permutation Matrix — Definition, Formula & Examples

A permutation matrix is a square matrix obtained by rearranging the rows (or columns) of the identity matrix. Each row and each column contains exactly one entry equal to 1, with all other entries equal to 0.

A permutation matrix $P$ of order $n$ is an $n \times n$ matrix in which every row and every column has precisely one nonzero entry, that entry being 1. Equivalently, $P$ is obtained from $I_n$ by applying a permutation $\sigma$ of $\{1, 2, \dots, n\}$ to its rows. Permutation matrices are orthogonal: $P^{-1} = P^T$ , and $\det(P) = \pm 1$ .

How It Works

Multiplying a matrix

A

on the left by a permutation matrix

P

rearranges the rows of

A

. Multiplying on the right by

P

rearranges the columns instead. Because

P

is orthogonal, its inverse is simply its transpose, so undoing the permutation costs nothing extra. Permutation matrices appear prominently in LU decomposition with partial pivoting, where the factorization takes the form

PA = LU

Worked Example

Problem: Let

P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}

and

A = \begin{pmatrix} 2 & 5 \\ 7 & 3 \\ 4 & 9 \end{pmatrix}

. Compute

PA

Identify the row permutation: Row 1 of

P

has its 1 in column 2, row 2 in column 3, and row 3 in column 1. So

P

sends row 2 of

A

to row 1, row 3 to row 2, and row 1 to row 3.

Multiply: Apply the permutation to the rows of

A

PA = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 2 & 5 \\ 7 & 3 \\ 4 & 9 \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ 4 & 9 \\ 2 & 5 \end{pmatrix}

Answer:

PA = \begin{pmatrix} 7 & 3 \\ 4 & 9 \\ 2 & 5 \end{pmatrix}

. The rows of

A

have been cyclically shifted upward.

Why It Matters

Permutation matrices are central to numerical linear algebra. In Gaussian elimination with partial pivoting, the algorithm records row swaps as a permutation matrix

P

so the factorization

PA = LU

remains stable. They also encode symmetries in group theory and combinatorics.

Common Mistakes

Mistake: Confusing left-multiplication and right-multiplication by a permutation matrix.

Correction:

PA

permutes the rows of

A

, while

AP

permutes the columns. The effects are different, so the side of multiplication matters.