/*! 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 55 What is the multiplicative ident... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 55

What is the multiplicative identity matrix?

Short Answer

Expert verified
The multiplicative identity matrix is a square matrix, which has ones on its principal diagonal (from top-left to bottom-right) and zeros everywhere else. When it multiplies any matrix, the resulting matrix is the original matrix.

Step by step solution

01

Understand Identity Matrix

An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. It's denoted by \(I_n\) or simply \(I\), where \(n\) is the order of the square matrix.
02

Structure of Identity Matrix

The identity matrix takes a simple form: it's a square matrix (which means it has the same number of rows as columns), with 1s on the diagonal and 0s everywhere else. For example, an identity matrix of order 3, \(I_3\) is represented as: \[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{bmatrix}\]
03

Properties of Identity Matrix

When identity matrix is multiplied with any matrix, the result is the original matrix itself. For example, if \(A\) is any matrix and \(I\) is an identity matrix, then \(A \times I = A\) and \(I \times A = A\). This is similar to multiplying numbers by 1; the product is the original number. The identity matrix \(I\) functions as a multiplicative identity for matrices.

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

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) a. If \(A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right],\) find \(A B\) b. Graph the object represented by matrix \(A B\). What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Evaluate: \(\left|\begin{array}{lllll}2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right|\)

Describe how to subtract matrices.

Describe the determinants \(D_{x}\) and \(D_{y}\) in terms of the coefficients and constants in a system of two equations in two variables.

will help you prepare for the material covered in the next section. Simplify the expression in each exercise. $$\frac{2(-5)-1(-4)}{5(-5)-6(-4)}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.