/*! 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 43 The dimensions of matrices \(A\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 43

The dimensions of matrices \(A\) and \(B\) are given. Find the dimensions of the product \(A B\) and of the product BA if the products are defined. If they are not defined, say so. $$A \text { is } 3 \times 5 ; B \text { is } 5 \times 2.$$

Short Answer

Expert verified
AB is 3x2; BA is not defined.

Step by step solution

01

Understanding Matrix Dimension Requirements for Multiplication

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. If this condition is met, the resulting matrix will have dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix.
02

Determine if Product AB is Defined

Matrix A has dimensions 3x5 and matrix B has dimensions 5x2. Since the number of columns in A (5) matches the number of rows in B (5), the product AB is defined. The dimensions of AB will be 3x2.
03

Calculate Dimensions of Product AB

Based on the multiplication condition being met, the resulting dimensions of the product AB will be determined by taking the number of rows from A and the number of columns from B. Therefore, AB will have dimensions 3x2.
04

Determine if Product BA is Defined

Matrix B has dimensions 5x2 and matrix A has dimensions 3x5. Since the number of columns in B (2) does not match the number of rows in A (3), the product BA is not defined. This mismatch means that BA cannot be calculated.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Dimensions
Matrices are rectangular grids of numbers arranged in rows and columns. When we talk about the dimensions of a matrix, we use the format \( m \times n \), where \( m \) is the number of rows, and \( n \) is the number of columns. This helps us understand the size and structure of a matrix.
  • If a matrix has dimensions 3x5, it means it has 3 rows and 5 columns.
  • This information is essential because it determines how we can perform operations such as addition or multiplication with other matrices.
Understanding matrix dimensions is crucial because it directly affects how we handle mathematical operations involving matrices.
Product of Matrices
The product of two matrices involves multiplying them together to form a new matrix. It is essential to first ensure that the matrices can be multiplied. The new matrix is created by following specific rules:
  • The number of columns in the first matrix must equal the number of rows in the second matrix.
  • The resulting product matrix will have dimensions based on the outer dimensions of the original matrices (rows from the first matrix and columns from the second).
For example, if matrix \( A \) is 3x5 and matrix \( B \) is 5x2, the product matrix \( AB \) will have dimensions 3x2. This process involves computing each element of the product matrix as the sum of products of corresponding elements from the rows of the first matrix and columns of the second.
Matrix Multiplication Conditions
There are specific conditions that must be met to multiply two matrices:
  • The number of columns in the first matrix must match the number of rows in the second matrix. This is the key condition to ensure that matrix multiplication is possible.
  • If this condition is not met, the product of the matrices cannot be defined.
For example, with matrices \( A \) (3x5) and \( B \) (5x2), the product \( AB \) is defined. However, if we try to multiply \( B \) (5x2) and \( A \) (3x5), we find the condition is not met. This is because the number of columns in \( B \) does not match the number of rows in \( A \), making the product \( BA \) undefined. Always ensure these conditions are checked before multiplying matrices to avoid errors.

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

As the price of a product increases, businesses usually increase the quantity manufactured. However, as the price increases, consumer demand-or the quantity of the product purchased by consumers-usually decreases. The price we see in the market place occurs when the quantity supplied and the quantity demanded are equal. This price is called the equilibrium price and this demand is called the equilibrium demand. The supply of a certain product is related to its price by the equation \(p=\frac{1}{3} q,\) where \(p\) is in dollars and \(q\) is the quantity supplied in hundreds of units. (a) If this product sells for 9 dollars, what quantity will be supplied by the manufacturer? (b) Suppose that consumer demand for the same product decreases as price increases according to the equation \(p=20-\frac{1}{5} q .\) If this product sells for 9 dollars, what quantity will consumers purchase? How does this compare with the quantity being supplied by the manufacturer at this price? (c) On the basis of parts (a) and (b), what should happen to the price? Explain. (d) Determine the equilibrium price at which the quantity supplied and quantity demanded are equal. What is the demand at this price?

For what value(s) of \(k\) will the following system of linear equations have no solution? infinitely many solutions? $$\begin{array}{r}x-2 y=3 \\\\-2 x+4 y=k\end{array}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \geq 3^{x}\\\ &y \geq 2 \end{aligned}$$

The current and estimated resident populations, \(y,\) (in percent) of Black and Spanish/Hispanic/Latino people in the United States for the years \(1990-2050\) are modeled by the following linear equations. $$\begin{aligned}&y=0.0515 x+12.3\\\&y=0.255 x+9.01\end{aligned}$$ In each case, \(x\) represents the number of years since 1990 . (Source: U.S. Census Bureau.) (a) Solve the system to find the year when these population percents were equal. (b) What percent of the U.S. resident population will be Spanish/Hispanic/Latino in the year found in part (a) (c) Graphically support the analytic solution in part (a). (d) Which population is increasing more rapidly?

Find a system of linear inequalities for which the graph is the region in the first quadrant between and inclusive of the pair of lines \(x+2 y-8=0\) and \(x+2 y=12\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

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.