6. For each of the following matrices, find all possible real values of (c, d) € R2 such that matrices C and D have same rank and then find the rank. d (a) C = (b) C = Го о 1 0] 0 0 0 с с 0 0 0 000 C 101 c 0 10 [1 0 2 1] (c) C= 0 d 0 000 D= D = 0, D= с d
6. For each of the following matrices, find all possible real values of (c, d) € R2 such that matrices C and D have same rank and then find the rank. d (a) C = (b) C = Го о 1 0] 0 0 0 с с 0 0 0 000 C 101 c 0 10 [1 0 2 1] (c) C= 0 d 0 000 D= D = 0, D= с d
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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Question
![**Linear Algebra Exercise: Exploring Matrix Rank**
**Problem Statement:**
For each of the following matrices, find all possible real values of \((c, d) \in \mathbb{R}^2\) such that matrices \(C\) and \(D\) have the same rank and then find the rank.
**Matrices:**
(a)
\[
C = \begin{bmatrix}
0 & 0 & 1 \\
c & 0 & 0 \\
0 & 0 & c
\end{bmatrix}, \quad
D = \begin{bmatrix}
d \\
d \\
d
\end{bmatrix}.
\]
(b)
\[
C = \begin{bmatrix}
1 & 0 & 1 & 2 \\
0 & c & 0 & 0
\end{bmatrix}, \quad
D = \begin{bmatrix}
d & 0 \\
0 & d
\end{bmatrix}.
\]
(c)
\[
C = \begin{bmatrix}
1 & 0 & 2 & 1 \\
0 & d & 0 & 0 \\
0 & 0 & c
\end{bmatrix}, \quad
D = \begin{bmatrix}
d & c \\
c & d
\end{bmatrix}.
\]
**Objective:**
1. Identify the rank of each matrix \(C\) and \(D\) to achieve parity.
2. Determine the conditions on \(c\) and \(d\) that equate the ranks of the respective matrices.
**Solution Approach:**
- Analyze each matrix pair to compute their ranks.
- Solve for parameters \(c\) and \(d\) to align the ranks of \(C\) and \(D\).
- Use determinants and row reduction as necessary to find these values efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fbf787395-3e33-4c3e-8905-237f2ad5e601%2Ffwqy0fra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Exercise: Exploring Matrix Rank**
**Problem Statement:**
For each of the following matrices, find all possible real values of \((c, d) \in \mathbb{R}^2\) such that matrices \(C\) and \(D\) have the same rank and then find the rank.
**Matrices:**
(a)
\[
C = \begin{bmatrix}
0 & 0 & 1 \\
c & 0 & 0 \\
0 & 0 & c
\end{bmatrix}, \quad
D = \begin{bmatrix}
d \\
d \\
d
\end{bmatrix}.
\]
(b)
\[
C = \begin{bmatrix}
1 & 0 & 1 & 2 \\
0 & c & 0 & 0
\end{bmatrix}, \quad
D = \begin{bmatrix}
d & 0 \\
0 & d
\end{bmatrix}.
\]
(c)
\[
C = \begin{bmatrix}
1 & 0 & 2 & 1 \\
0 & d & 0 & 0 \\
0 & 0 & c
\end{bmatrix}, \quad
D = \begin{bmatrix}
d & c \\
c & d
\end{bmatrix}.
\]
**Objective:**
1. Identify the rank of each matrix \(C\) and \(D\) to achieve parity.
2. Determine the conditions on \(c\) and \(d\) that equate the ranks of the respective matrices.
**Solution Approach:**
- Analyze each matrix pair to compute their ranks.
- Solve for parameters \(c\) and \(d\) to align the ranks of \(C\) and \(D\).
- Use determinants and row reduction as necessary to find these values efficiently.
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