rank(A) = nullity(A) = rank(A) + nullity(A) choose
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
![**Find the rank and the nullity of the matrix**
Matrix \( A \) is given as:
\[
A = \begin{bmatrix}
2 & -8 & 0 & -3 & -9 \\
-1 & 3 & 0 & -2 & 3 \\
1 & -4 & 0 & -1 & -4
\end{bmatrix}
\]
**Explanation**
To find the rank of the matrix, one can perform row reduction to bring the matrix to its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF), counting the number of non-zero rows. The rank is the dimension of the row space or column space of the matrix.
The nullity of the matrix is given by the dimension of the null space, which can be found by subtracting the rank from the number of columns in the matrix. In this case, the matrix \( A \) has 5 columns.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e989a36-a976-4f9a-8f93-2801096d9bdd%2F79fe58c4-918f-4500-9dd4-348700329dd3%2F56za794_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the rank and the nullity of the matrix**
Matrix \( A \) is given as:
\[
A = \begin{bmatrix}
2 & -8 & 0 & -3 & -9 \\
-1 & 3 & 0 & -2 & 3 \\
1 & -4 & 0 & -1 & -4
\end{bmatrix}
\]
**Explanation**
To find the rank of the matrix, one can perform row reduction to bring the matrix to its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF), counting the number of non-zero rows. The rank is the dimension of the row space or column space of the matrix.
The nullity of the matrix is given by the dimension of the null space, which can be found by subtracting the rank from the number of columns in the matrix. In this case, the matrix \( A \) has 5 columns.
![The image represents a mathematical exercise related to linear algebra. It includes three expressions that require input from the user:
1. **rank(A) =** [Text box]
- Users are prompted to input the rank of matrix A.
2. **nullity(A) =** [Text box]
- Users are prompted to input the nullity of matrix A.
3. **rank(A) + nullity(A) =** [Dropdown menu labeled "choose"]
- Users are provided with a dropdown menu to select the result that represents the sum of the rank and nullity of matrix A.
This exercise is likely part of a lesson on the Rank-Nullity Theorem, which states that for a given matrix A, the sum of its rank and nullity equals the number of columns in the matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e989a36-a976-4f9a-8f93-2801096d9bdd%2F79fe58c4-918f-4500-9dd4-348700329dd3%2Focoz9lan_processed.png&w=3840&q=75)
Transcribed Image Text:The image represents a mathematical exercise related to linear algebra. It includes three expressions that require input from the user:
1. **rank(A) =** [Text box]
- Users are prompted to input the rank of matrix A.
2. **nullity(A) =** [Text box]
- Users are prompted to input the nullity of matrix A.
3. **rank(A) + nullity(A) =** [Dropdown menu labeled "choose"]
- Users are provided with a dropdown menu to select the result that represents the sum of the rank and nullity of matrix A.
This exercise is likely part of a lesson on the Rank-Nullity Theorem, which states that for a given matrix A, the sum of its rank and nullity equals the number of columns in the matrix.
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