4. Consider the matrices A and B given below rank and nullity of A. A || 2 2 -2 -1 0 -1 1 0 CO TO - 2 -2 -6 12 0 4 -1 -2-4 - 60-9 3 15 B = State the 0 1] 1 0-2 3 0 1 -30 1 01 0 12 000 000 000
4. Consider the matrices A and B given below rank and nullity of A. A || 2 2 -2 -1 0 -1 1 0 CO TO - 2 -2 -6 12 0 4 -1 -2-4 - 60-9 3 15 B = State the 0 1] 1 0-2 3 0 1 -30 1 01 0 12 000 000 000
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
![**Problem 4: Matrix Analysis**
Consider the matrices \( A \) and \( B \) given below. State the rank and nullity of \( A \).
Matrix \( A \):
\[
A =
\begin{bmatrix}
2 & 2 & -2 & 0 & 0 & 4 \\
-1 & -1 & 1 & 0 & -1 & -4 \\
2 & -2 & -6 & 12 & -2 & -4 \\
3 & 6 & 0 & -9 & 3 & 15
\end{bmatrix}
\]
Matrix \( B \):
\[
B =
\begin{bmatrix}
1 & 0 & -2 & 3 & 0 & 1 \\
0 & 1 & 1 & -3 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
### Explanation
- **Rank of a Matrix**: The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors) in the matrix. It can be determined by bringing the matrix to its row-echelon form and counting the number of non-zero rows.
- **Nullity of a Matrix**: The nullity is the dimension of the solution space to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \). It is calculated as the number of columns minus the rank of the matrix.
These concepts are important in linear algebra for understanding the solution set of linear systems and the transformations represented by the matrix.
*Note: To solve the problem, one would perform Gaussian elimination on matrix \( A \) to find its rank and nullity.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30f79184-3047-455a-b3d8-f2ad47623cdf%2Faf484355-d5f4-4c28-9ac0-f5a3a22182d4%2Fisiqocp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4: Matrix Analysis**
Consider the matrices \( A \) and \( B \) given below. State the rank and nullity of \( A \).
Matrix \( A \):
\[
A =
\begin{bmatrix}
2 & 2 & -2 & 0 & 0 & 4 \\
-1 & -1 & 1 & 0 & -1 & -4 \\
2 & -2 & -6 & 12 & -2 & -4 \\
3 & 6 & 0 & -9 & 3 & 15
\end{bmatrix}
\]
Matrix \( B \):
\[
B =
\begin{bmatrix}
1 & 0 & -2 & 3 & 0 & 1 \\
0 & 1 & 1 & -3 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
### Explanation
- **Rank of a Matrix**: The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors) in the matrix. It can be determined by bringing the matrix to its row-echelon form and counting the number of non-zero rows.
- **Nullity of a Matrix**: The nullity is the dimension of the solution space to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \). It is calculated as the number of columns minus the rank of the matrix.
These concepts are important in linear algebra for understanding the solution set of linear systems and the transformations represented by the matrix.
*Note: To solve the problem, one would perform Gaussian elimination on matrix \( A \) to find its rank and nullity.*
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