(a) Write down the inverses of each of the 3 ×3 permutation matrices (1.30). (b) Which ones are

step 1 So for chapter 1, section 5, question 14, we have p equals the identity matrix, which is 1 -1. I

(a) Write down the inverses of each of the 3 ×3 permutation...

(a) Write down the inverses of each of the $3 \times 3$ permutation matrices (1.30). (b) Which ones are their own inverses, $P^{-1}=P$ ? (c) Can you find a non-elementary permutation matrix $P$ that is its own inverse: $P^{-1}=P$ ?

Key Concepts

Permutation Matrix

A permutation matrix is a square matrix obtained by permuting the rows (or columns) of an identity matrix. Each row and column contains exactly one entry equal to 1 and all other entries are 0, which makes these matrices representations of permutations in linear algebra. They are used to rearrange the order of the basis vectors in a systematic, invertible manner.

Inverse of a Permutation Matrix

The inverse of a permutation matrix is another permutation matrix that reverts the permutation applied by the original matrix. Since permutation matrices essentially represent permutation functions, their inverses undo the row or column reordering. In most cases, the inverse of a permutation matrix is its transpose, as the operation of transposing effectively reverses the initial permutation.

Self-Inverse (Involutory) Matrix

A self-inverse or involutory matrix is one that is its own inverse, meaning when the matrix is multiplied by itself, the result is the identity matrix. For permutation matrices, this property corresponds to those permutations that are their own inverses. In permutation group terms, a permutation is an involution if its cycle decomposition consists solely of 1-cycles and transpositions.

Elementary and Non-elementary Permutations

Elementary permutation matrices typically perform a simple operation such as swapping two rows or columns, corresponding to a single transposition. These matrices are inherently self-inverse because performing the swap twice restores the original order. Non-elementary permutation matrices, on the other hand, represent more complex permutations that might involve longer cycles. However, a non-elementary permutation matrix can still be self-inverse if the overall permutation decomposes into disjoint transpositions, even if it is not an immediate swap of two entries.

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