Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Determine Whether the Transpose of a Matrix Product Equals the Product of Transposes**
In the study of matrices, particularly in linear algebra, we often come across operations involving transposing matrices. One such operation is the transpose of the product of matrices. A common question is to determine if the transpose of the product of matrices equals the product of their transposes, but in reverse order.
**Question**:
Determine whether or not \((ABC)^T = C^T B^T A^T\) for matrices \(A\), \(B\), and \(C\).
**Explanation**:
- **(ABC)ᵀ**: This denotes the transpose of the product of matrices \(A\), \(B\), and \(C\).
- **CᵀBᵀAᵀ**: This denotes the product of the transposes of matrices \(C\), \(B\), and \(A\), but in reverse order.
The question asks whether these two expressions are equivalent for general matrices \(A\), \(B\), and \(C\).
**Key Concept**:
For any matrices \(A\), \(B\), and \(C\) where the products are defined, the transpose of a product of matrices satisfies the property:
\[
(ABC)^T = C^T B^T A^T
\]
Thus, the statement is true under the assumption that all matrices and their products are appropriately defined (i.e., the matrices are conformable for the operations).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20fbb71c-9fd5-4b59-9900-a3864652c523%2F7699fc9f-50ee-42ff-b64c-2d5588fde7f4%2Frsbf6jg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine Whether the Transpose of a Matrix Product Equals the Product of Transposes**
In the study of matrices, particularly in linear algebra, we often come across operations involving transposing matrices. One such operation is the transpose of the product of matrices. A common question is to determine if the transpose of the product of matrices equals the product of their transposes, but in reverse order.
**Question**:
Determine whether or not \((ABC)^T = C^T B^T A^T\) for matrices \(A\), \(B\), and \(C\).
**Explanation**:
- **(ABC)ᵀ**: This denotes the transpose of the product of matrices \(A\), \(B\), and \(C\).
- **CᵀBᵀAᵀ**: This denotes the product of the transposes of matrices \(C\), \(B\), and \(A\), but in reverse order.
The question asks whether these two expressions are equivalent for general matrices \(A\), \(B\), and \(C\).
**Key Concept**:
For any matrices \(A\), \(B\), and \(C\) where the products are defined, the transpose of a product of matrices satisfies the property:
\[
(ABC)^T = C^T B^T A^T
\]
Thus, the statement is true under the assumption that all matrices and their products are appropriately defined (i.e., the matrices are conformable for the operations).
Expert Solution
Step 1
Given-
To Determine- Whether the above statement is correct or not for matrices A,B,C.
Theorem Used- Transposing of product of the matrices P and Q is given by,
.
Step by step
Solved in 2 steps
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