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If a matrix A is 4 x 7 and the product AB is 4×4, what is the size of B? The size of B is X

If a matrix A is 4 x 7 and the product AB is 4×4, what is the size of B? The size of B is X

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Matrix Size Determination Problem #### Problem Statement: If a matrix \( A \) is \( 4 \times 7 \) and the product \( AB \) is \( 4 \times 4 \), what is the size of \( B \)? #### Solution: To determine the size of matrix \( B \), we need to understand the rules of matrix multiplication. The product of two matrices \( A \) and \( B \) is defined if and only if the number of columns in \( A \) is equal to the number of rows in \( B \). 1. Matrix \( A \) is \( 4 \times 7 \) (4 rows and 7 columns). 2. Matrix \( AB \) is \( 4 \times 4 \). Given this information, let matrix \( B \) be of size \( 7 \times k \): - The number of columns in \( A \) (which is 7) must equal the number of rows in \( B \). - The number of rows in the product matrix \( AB \) (which is 4) must equal the number of rows in \( A \), which confirms our calculation. So, matrix \( B \) must have the same number of columns as \( AB \), which is 4. Thus, the size of matrix \( B \) is: \[ \boxed{7} \times \boxed{4} \] Input the values in the provided boxes. This understanding helps reinforce the concept of matrix multiplication dimensions and ensures the students comprehend the relationship between the dimensions of matrices involved in multiplication.
Transcribed Image Text:### Matrix Size Determination Problem #### Problem Statement: If a matrix \( A \) is \( 4 \times 7 \) and the product \( AB \) is \( 4 \times 4 \), what is the size of \( B \)? #### Solution: To determine the size of matrix \( B \), we need to understand the rules of matrix multiplication. The product of two matrices \( A \) and \( B \) is defined if and only if the number of columns in \( A \) is equal to the number of rows in \( B \). 1. Matrix \( A \) is \( 4 \times 7 \) (4 rows and 7 columns). 2. Matrix \( AB \) is \( 4 \times 4 \). Given this information, let matrix \( B \) be of size \( 7 \times k \): - The number of columns in \( A \) (which is 7) must equal the number of rows in \( B \). - The number of rows in the product matrix \( AB \) (which is 4) must equal the number of rows in \( A \), which confirms our calculation. So, matrix \( B \) must have the same number of columns as \( AB \), which is 4. Thus, the size of matrix \( B \) is: \[ \boxed{7} \times \boxed{4} \] Input the values in the provided boxes. This understanding helps reinforce the concept of matrix multiplication dimensions and ensures the students comprehend the relationship between the dimensions of matrices involved in multiplication.
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