2. Find the rank of the following matrices. /1 1 0 GID 0 1 1 (b) 2 1 1 1 1 0 (a) 1) (4) (272) (f) 1234 -4 2 4 1 62 -8 1 -3 1, 0 1 1) 3 0 51 (e) (g) 12 3 1 1 0 1 2 02-30 1 (Ⓒ) (₁² Gi 14 4 2 12 10 00 1 0 1 2202 1101 1 1 0 1 02
2. Find the rank of the following matrices. /1 1 0 GID 0 1 1 (b) 2 1 1 1 1 0 (a) 1) (4) (272) (f) 1234 -4 2 4 1 62 -8 1 -3 1, 0 1 1) 3 0 51 (e) (g) 12 3 1 1 0 1 2 02-30 1 (Ⓒ) (₁² Gi 14 4 2 12 10 00 1 0 1 2202 1101 1 1 0 1 02
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Matrix Rank Problems and Operations
#### Problem 2: Rank of Matrices
Determine the rank of each of the following matrices:
(a)
\[
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0
\end{bmatrix}
\]
(b)
\[
\begin{bmatrix}
1 & 1 & 0 \\
2 & 1 & 1 \\
1 & 1 & 1
\end{bmatrix}
\]
(c)
\[
\begin{bmatrix}
1 & 0 & 2 \\
1 & 1 & 4
\end{bmatrix}
\]
(d)
\[
\begin{bmatrix}
1 & 2 & 1 \\
2 & 4 & 2
\end{bmatrix}
\]
(e)
\[
\begin{bmatrix}
1 & 2 & 3 & 1 & 1 \\
1 & 4 & 0 & 1 & 2 \\
0 & 2 & -3 & 0 & 1 \\
1 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
(f)
\[
\begin{bmatrix}
1 & 2 & 0 & 1 & 1 \\
2 & 4 & 1 & 3 & 0 \\
3 & 6 & 2 & 5 & 1 \\
-4 & -8 & 1 & -3 & 1
\end{bmatrix}
\]
(g)
\[
\begin{bmatrix}
1 & 1 & 0 & 1 \\
2 & 2 & 0 & 2 \\
1 & 1 & 0 & 1 \\
1 & 1 & 0 & 1
\end{bmatrix}
\]
#### Problem 3: Zero Matrix and Rank
Show that for any \(m \times n\) matrix \(A\), \(\text{rank}(A) = 0\) if and only if \(A\) is the zero matrix.
#### Problem 4: Row and Column Operations
Use elementary row and column operations to transform each of the following matrices into a matrix \(D\) satisfying the conditions of Theorem](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F905809b3-f86f-4a48-848e-914423a7f462%2F5b9acc2b-d44d-452f-8630-5fb63fa403c4%2Ft2xe6c2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Rank Problems and Operations
#### Problem 2: Rank of Matrices
Determine the rank of each of the following matrices:
(a)
\[
\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 1 & 0
\end{bmatrix}
\]
(b)
\[
\begin{bmatrix}
1 & 1 & 0 \\
2 & 1 & 1 \\
1 & 1 & 1
\end{bmatrix}
\]
(c)
\[
\begin{bmatrix}
1 & 0 & 2 \\
1 & 1 & 4
\end{bmatrix}
\]
(d)
\[
\begin{bmatrix}
1 & 2 & 1 \\
2 & 4 & 2
\end{bmatrix}
\]
(e)
\[
\begin{bmatrix}
1 & 2 & 3 & 1 & 1 \\
1 & 4 & 0 & 1 & 2 \\
0 & 2 & -3 & 0 & 1 \\
1 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
(f)
\[
\begin{bmatrix}
1 & 2 & 0 & 1 & 1 \\
2 & 4 & 1 & 3 & 0 \\
3 & 6 & 2 & 5 & 1 \\
-4 & -8 & 1 & -3 & 1
\end{bmatrix}
\]
(g)
\[
\begin{bmatrix}
1 & 1 & 0 & 1 \\
2 & 2 & 0 & 2 \\
1 & 1 & 0 & 1 \\
1 & 1 & 0 & 1
\end{bmatrix}
\]
#### Problem 3: Zero Matrix and Rank
Show that for any \(m \times n\) matrix \(A\), \(\text{rank}(A) = 0\) if and only if \(A\) is the zero matrix.
#### Problem 4: Row and Column Operations
Use elementary row and column operations to transform each of the following matrices into a matrix \(D\) satisfying the conditions of Theorem
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