3.6. Let A be the 3 x 3 matrix all of whose entries are 1. Find a matrix B such that for any 3 x n matrix X. AX 2X BX
3.6. Let A be the 3 x 3 matrix all of whose entries are 1. Find a matrix B such that for any 3 x n matrix X. AX 2X BX
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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![### Problem 3.6
**Problem Statement:**
Let \( A \) be the \( 3 \times 3 \) matrix all of whose entries are 1. Find a matrix \( B \) such that
\[
AX - 2X = BX
\]
for any \( 3 \times n \) matrix \( X \).
**Detailed Explanation:**
1. **Matrix A**: This is a \( 3 \times 3 \) matrix where every entry is 1.
\[
A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{pmatrix}
\]
2. **Matrix X**: This is a \( 3 \times n \) matrix, where \( n \) is any positive integer.
3. **Equation to Satisfy**:
\[
AX - 2X = BX
\]
We need to find a \( 3 \times 3 \) matrix \( B \) that satisfies this equation for any \( 3 \times n \) matrix \( X \).
4. **Step-by-Step Solution**:
- **Calculate \( AX \)**: When multiplying \( A \) by \( X \), since all entries in \( A \) are 1, each element of the resulting matrix \( AX \) is the sum of the corresponding column elements of \( X \).
For example, consider \( X \) as:
\[
X = \begin{pmatrix}
x_{11} & x_{12} & \dots & x_{1n} \\
x_{21} & x_{22} & \dots & x_{2n} \\
x_{31} & x_{32} & \dots & x_{3n}
\end{pmatrix}
\]
Then:
\[
AX = \begin{pmatrix}
x_{11} + x_{21} + x_{31} & x_{12} + x_{22} + x_{32} & \dots & x_{1n} + x_{2n} + x_{3n} \\
x_{11} + x_{21} + x_{31} & x_{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ee3a518-8279-4c7a-b55b-ef9ad8090828%2F81c1b20e-7396-4e77-8c78-b70513c9ef95%2Fuagwamd_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3.6
**Problem Statement:**
Let \( A \) be the \( 3 \times 3 \) matrix all of whose entries are 1. Find a matrix \( B \) such that
\[
AX - 2X = BX
\]
for any \( 3 \times n \) matrix \( X \).
**Detailed Explanation:**
1. **Matrix A**: This is a \( 3 \times 3 \) matrix where every entry is 1.
\[
A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{pmatrix}
\]
2. **Matrix X**: This is a \( 3 \times n \) matrix, where \( n \) is any positive integer.
3. **Equation to Satisfy**:
\[
AX - 2X = BX
\]
We need to find a \( 3 \times 3 \) matrix \( B \) that satisfies this equation for any \( 3 \times n \) matrix \( X \).
4. **Step-by-Step Solution**:
- **Calculate \( AX \)**: When multiplying \( A \) by \( X \), since all entries in \( A \) are 1, each element of the resulting matrix \( AX \) is the sum of the corresponding column elements of \( X \).
For example, consider \( X \) as:
\[
X = \begin{pmatrix}
x_{11} & x_{12} & \dots & x_{1n} \\
x_{21} & x_{22} & \dots & x_{2n} \\
x_{31} & x_{32} & \dots & x_{3n}
\end{pmatrix}
\]
Then:
\[
AX = \begin{pmatrix}
x_{11} + x_{21} + x_{31} & x_{12} + x_{22} + x_{32} & \dots & x_{1n} + x_{2n} + x_{3n} \\
x_{11} + x_{21} + x_{31} & x_{
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