### Matrix Reduction and Null Space Basis#### Matrix AThe matrix $ A $ is given by:\[A = \begin{bmatrix} -2 & -2 & 2 & 8 \\ -6 & -2 & 1 & 1 \\ -6 & 0 & -1 & -9 \end{bmatrix}\]#### Row ReductionThe matrix $ A $ reduces to its row-echelon form:\[\begin{bmatrix} -2 & 0 & 0 & -2 \\ 0 & 2 & 0 & -4 \\ 0 & 0 & 1 & 3 \end{bmatrix}\]#### Null SpaceTo find a basis for the null space of matrix $ A $ (Nul $ A $), follow these steps:1. **Identify Free Variables**: In the row-echelon form, any column without a leading entry is a free variable.2. **Express Basic Variables in Terms of Free Variables**: Solve the system for the basic variables in terms of the free variables.#### Basis for Nul AThe null space basis is represented as a set of vectors. The placeholders are provided to input the solutions:\[\begin{bmatrix} \text{[ \ \ ]} \\ \text{[ \ \ ]} \\ \text{[ \ \ ]} \\ \text{[ \ \ ]} \end{bmatrix}\]

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2 - 2 8 - 20 0 -2 A matrix A= - 2 1 1 reduces to 2 0 – 4 6 1 -9 0 1 3 Find a basis for Nul A: 2.

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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### Matrix Reduction and Null Space Basis

#### Matrix A
The matrix $ A $ is given by:

\[
A = \begin{bmatrix}
-2 & -2 & 2 & 8 \\
-6 & -2 & 1 & 1 \\
-6 & 0 & -1 & -9
\end{bmatrix}
\]

#### Row Reduction
The matrix $ A $ reduces to its row-echelon form:

\[
\begin{bmatrix}
-2 & 0 & 0 & -2 \\
0 & 2 & 0 & -4 \\
0 & 0 & 1 & 3
\end{bmatrix}
\]

#### Null Space
To find a basis for the null space of matrix $ A $ (Nul $ A $), follow these steps:

1. **Identify Free Variables**: In the row-echelon form, any column without a leading entry is a free variable.
2. **Express Basic Variables in Terms of Free Variables**: Solve the system for the basic variables in terms of the free variables.

#### Basis for Nul A
The null space basis is represented as a set of vectors. The placeholders are provided to input the solutions:

\[
\begin{bmatrix}
\text{[ \ \ ]} \\
\text{[ \ \ ]} \\
\text{[ \ \ ]} \\
\text{[ \ \ ]}
\end{bmatrix}
\]

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