/*! 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 57 Explain why a matrix that does n... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 57

Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.

Short Answer

Expert verified
A matrix must be square in order to have an inverse because it needs to have a non-zero determinant, which can only be computed for square matrices, and it needs to be able to multiply with its inverse to form the identity matrix, which requires the number of rows of the first to equal the number of columns in the second.

Step by step solution

01

Understand the inverse of a matrix

The inverse of a matrix A is another matrix, denoted as A^-1, such that when A is multiplied with A^-1, the result is the identity matrix. The identity matrix is a special square matrix with 1's on the diagonal and 0's elsewhere.
02

Express the requirement for an inverse matrix

For a matrix to have an inverse, it must satisfy two conditions: 1) It must be square (i.e., have the same number of rows and columns). 2) Its determinant must be non-zero. A non-square matrix will not have a determinant, and thus an inverse cannot be defined for it.
03

Connect the square requirement to multiplication

Matrix multiplication is only possible if the number of columns of the first matrix is equal to the number of rows of the second one. Hence, when attempting to multiply a matrix by its inverse to get the identity matrix (which is always square), both the original matrix and its inverse must be square matrices. Otherwise, the multiplication would not be possible.

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 \(B ?\)

Explain how to find the multiplicative inverse for a \(3 \times 3\) invertible matrix.

Let $$\begin{aligned}&A=\left[\begin{array}{ll}1 & 0 \\\0 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 0 \\\0 & -1\end{array}\right], \quad C=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\\\&D=\left[\begin{array}{rr}-1 & 0 \\\0 & -1\end{array}\right]\end{aligned}$$ Use any three of the matrices to verify a distributive property.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give an example of a \(2 \times 2\) matrix that is its own inverse.

Determinants are used to write an equation of a line passing. through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$\left|\begin{array}{lll}x & y & 1 \\\x_{1} & y_{1} & 1 \\\x_{2} & y_{2} & 1\end{array}\right|=0$$ Use the determinant to write an equation of the line passing through \((3,-5)\) and \((-2,6) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

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.