Let A and B be n× n matrices. Show that if the ith row of A has all zero entries, then the ith row of AB will have all zero entries. Let A = [a¡] and B = [b¡] be two diagonal n×n matrices. Then the ijth entry of the product AB is Cij = k = 1 If the ith row of A has all zero entries, evaluate the entries ajk for all k = 1, 2, .., n. ajk = 1+n Evaluate the entries C for all j = 1, 2, ..., n. Cij = AB Thus, if the ith row of A has all zero entries, then the ith row of AB has all zero entries.
Let A and B be n× n matrices. Show that if the ith row of A has all zero entries, then the ith row of AB will have all zero entries. Let A = [a¡] and B = [b¡] be two diagonal n×n matrices. Then the ijth entry of the product AB is Cij = k = 1 If the ith row of A has all zero entries, evaluate the entries ajk for all k = 1, 2, .., n. ajk = 1+n Evaluate the entries C for all j = 1, 2, ..., n. Cij = AB Thus, if the ith row of A has all zero entries, then the ith row of AB has all zero entries.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Matrix Row Zero Product Theorem**
Let \( A \) and \( B \) be \( n \times n \) matrices. We aim to show that if the \( i \)th row of \( A \) has all zero entries, then the \( i \)th row of the product \( AB \) will also have all zero entries.
Consider:
- Let \( A = [a_{ij}] \) and \( B = [b_{ij}] \) be two diagonal \( n \times n \) matrices. The \( i \)th entry of the product \( AB \) is given by:
\[
c_{ij} = \sum_{k=1}^{n} 0
\]
- If the \( i \)th row of \( A \) has all zero entries, evaluate the entries \( a_{ik} \) for all \( k = 1, 2, \ldots, n \).
\[
a_{ik} = 1 + n
\]
- Evaluate the entries \( c_{ij} \) for all \( j = 1, 2, \ldots, n \).
\[
c_{ij} = AB
\]
Thus, if the \( i \)th row of \( A \) has all zero entries, the \( i \)th row of \( AB \) will have all zero entries.
**Example to Show the Converse is Not True**
To demonstrate the converse is not true, use \( 2 \times 2 \) matrices as an example:
\[
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 14 & 18 \end{bmatrix}
\]
Matrix \( B \) is:
\[
B = \begin{bmatrix} 14 & 18 \\ -7 & -9 \end{bmatrix}
\]
In this example, notice the incorrect statement marked with a red cross indicates mistakes in the attempted computation or a potential misapplication of the theorem. The green check marks indicate corrections or validations.
This exercise clarifies that while the zero row product rule holds true, its converse does not necessarily apply.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a7c37e1-d117-46e5-8cc2-cb36983cacd6%2F6dff7dea-9a06-463c-bb72-cb5058e187b9%2Fhqat4cn_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Row Zero Product Theorem**
Let \( A \) and \( B \) be \( n \times n \) matrices. We aim to show that if the \( i \)th row of \( A \) has all zero entries, then the \( i \)th row of the product \( AB \) will also have all zero entries.
Consider:
- Let \( A = [a_{ij}] \) and \( B = [b_{ij}] \) be two diagonal \( n \times n \) matrices. The \( i \)th entry of the product \( AB \) is given by:
\[
c_{ij} = \sum_{k=1}^{n} 0
\]
- If the \( i \)th row of \( A \) has all zero entries, evaluate the entries \( a_{ik} \) for all \( k = 1, 2, \ldots, n \).
\[
a_{ik} = 1 + n
\]
- Evaluate the entries \( c_{ij} \) for all \( j = 1, 2, \ldots, n \).
\[
c_{ij} = AB
\]
Thus, if the \( i \)th row of \( A \) has all zero entries, the \( i \)th row of \( AB \) will have all zero entries.
**Example to Show the Converse is Not True**
To demonstrate the converse is not true, use \( 2 \times 2 \) matrices as an example:
\[
A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 14 & 18 \end{bmatrix}
\]
Matrix \( B \) is:
\[
B = \begin{bmatrix} 14 & 18 \\ -7 & -9 \end{bmatrix}
\]
In this example, notice the incorrect statement marked with a red cross indicates mistakes in the attempted computation or a potential misapplication of the theorem. The green check marks indicate corrections or validations.
This exercise clarifies that while the zero row product rule holds true, its converse does not necessarily apply.
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