/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 How many \(n \times n\) permutat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 17

How many \(n \times n\) permutation matrices are there?

Short Answer

Expert verified
The total number of \(n \times n\) permutation matrices is equal to the factorial of n, or \(n!\).

Step by step solution

01

Define an n 脳 n permutation matrix

A permutation matrix is a square matrix (same number of rows and columns) where each row and each column has exactly one 1 and the remaining entries are 0. It arises from the identity matrix with its rows and columns permuted.
02

Analyze the first row

For an n 脳 n permutation matrix, there are n possible positions for the 1 in the first row. Since there can only be one 1 in each row and column, there are (n-1) possible positions for the 1 in the second row, (n-2) possible positions for the 1 in the third row, and so on.
03

Apply the factorial function to the problem

As there are n possibilities for the first row, (n-1) possibilities for the second row, (n-2) possibilities for the third row, and so on, until only one possibility remains for the last row, we can find the total number of permutations by multiplying all of these possibilities. This product is known as the factorial function, denoted by n!: \[n! = n \times (n-1) \times (n-2) \times \ldots \times 1\]
04

Conclusion

The total number of n 脳 n permutation matrices is equal to the factorial of n, or n!.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use equation (1) with weight vector \(\mathbf{w}=\) \(\left(\frac{1}{4}, \frac{1}{2}, \frac{1}{4}\right)^{T}\) to define an inner product for \(\mathbb{R}^{3},\) and let \(\mathbf{x}=(1,1,1)^{T}\) and \(\mathbf{y}=(-5,1,3)^{T}\) (a) Show that \(x\) and \(y\) are orthogonal with respect to this inner product. (b) Compute the values of \(\|\mathbf{x}\|\) and \(\|\mathbf{y}\|\) with respect to this inner product.

Let \(T_{n}(x)\) denote the Chebyshev polynomial of degree \(n,\) and define $$U_{n-1}(x)=\frac{1}{n} T_{n}^{\prime}(x)$$ for \(n=1,2, \ldots\) (a) Compute \(U_{0}(x), U_{1}(x),\) and \(U_{2}(x)\) (b) Show that if \(x=\cos \theta,\) then $$U_{n-1}(x)=\frac{\sin n \theta}{\sin \theta}$$

Let \(p_{0}, p_{1}, \ldots\) be a sequence of orthogonal polynomials and let \(a_{n}\) denote the lead coefficient of \(p_{n}\) Prove that $$\left\|p_{n}\right\|^{2}=a_{n}\left\langle x^{n}, p_{n}\right\rangle$$

Let \(P=A\left(A^{T} A\right)^{-1} A^{T},\) where \(A\) is an \(m \times n\) matrix of rank \(n\) (a) Show that \(P^{2}=P\) (b) Prove that \(P^{k}=P\) for \(k=1,2, \ldots\) (c) Show that \(P\) is symmetric. [Hint: If \(B\) is nonsingular, then \(\left.\left(B^{-1}\right)^{T}=\left(B^{T}\right)^{-1} .\right]\)

Let \(A\) be an \(m \times n\) matrix, let \(P\) be the projection matrix that projects vectors in \(\mathbb{R}^{m}\) onto \(R(A),\) and let \(Q\) be the projection matrix that projects vectors in \(\mathbb{R}^{n}\) onto \(R\left(A^{T}\right) .\) Show that (a) \(I-P\) is the projection matrix from \(\mathbb{R}^{m}\) onto \(N\left(A^{T}\right)\) (b) \(I-Q\) is the projection matrix from \(\mathbb{R}^{n}\) onto \(N(A)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.