Find the inverse of the following permutation matrices: (a)...

Find the inverse of the following permutation matrices: (a) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$, (b) $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$ (c) $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$ (d) $\left(\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\end{array}\right)$

Find the inverse of the following permutation matrices:
(a) $\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\end{array}\right)$,
(b) $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\end{array}\right)$
(c) $\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$
(d) $\left(\begin{array}{lllll}1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0\end{array}\right)$

Key Concepts

Permutation Matrix

A permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column, with all other entries being 0. Such matrices represent permutations of the basis vectors when used to multiply another matrix. They play a central role in many areas of linear algebra and combinatorics, serving as a way to encode and operate on permutations in a matrix format.

Matrix Inverse

The inverse of a matrix A is a matrix A?¹ such that AA?¹ = A?¹A = I, where I is the identity matrix. In the context of permutation matrices, finding the inverse is particularly streamlined because the inverse of a permutation matrix is also a permutation matrix, often obtained by simply reversing the permutation encoded by the original matrix.

Transpose of a Matrix

The transpose of a matrix is formed by interchanging its rows and columns. For any permutation matrix, the transpose is not only easy to compute but also serves as its inverse. This is due to the orthogonality of permutation matrices, which guarantees that multiplying a permutation matrix by its transpose yields the identity matrix, confirming that the transpose is indeed the inverse.

Transcript

00:01 Number 435, they want us to find the inverse of this matrix here on the left.

00:09 Okay, and the way we're going to do that is to write the identity matrix here on the right and do elementary row operations until we get the identity matrix here on the left.

00:22 Okay, so this is what we were given.

00:25 This is a six -by -six identity matrix.

00:28 Okay, and we almost have the identity.

00:30 Matrix here, but we have to get rid of these five ones and row six.

00:38 Okay.

00:40 And we can do that by subtracting each of these rows from row six.

00:47 Okay, so we don't really need to keep rewriting the whole thing because we're only going to be working with row six.

00:58 So i'm going to do row six minus row five.

01:05 And then i'm going to do row six minus row four, row six minus row three, row six minus row two, and row six minus row one.

01:27 Okay.

01:32 So let's start with row six minus row five.

01:36 Okay.