Write solution in parametric form from vector ### Understanding Matrix and Vector OperationsConsider the following matrix **A** and vectors **p** and **q**:\[A = \begin{bmatrix}1 & -3 \\-3 & 0 \\5 & 2\end{bmatrix}\quadp = \begin{bmatrix}11 \\-6 \\4\end{bmatrix}\quadq = \begin{bmatrix}9 \\-9 \\11\end{bmatrix}\]#### Matrix **A**- **A** is a 3x2 matrix, meaning it has 3 rows and 2 columns.- The entries of the matrix are as follows: - First row: 1, -3 - Second row: -3, 0 - Third row: 5, 2#### Vector **p**- **p** is a 3x1 column vector.- The entries of vector **p** are: - First element: 11 - Second element: -6 - Third element: 4#### Vector **q**- **q** is also a 3x1 column vector.- The entries of vector **q** are: - First element: 9 - Second element: -9 - Third element: 11These matrices and vectors can be used in a variety of linear algebra operations such as matrix multiplication, finding the dot product, and solving systems of linear equations. Understanding their structure is fundamental in linear algebra and various applications in mathematics, physics, and engineering. **Solving Systems of Linear Equations**In this section, we will tackle the problem presented below:**Problem Statement:** Solve the equation $Ax = p$, or show that the equation is inconsistent. Write your solution in parametric vector form.**Explanation and Steps:**1. **Identify the Given Equation:** The given equation is $Ax = p$, where $A$ is a matrix, $x$ is the vector of variables, and $p$ is the resultant vector.2. **Evaluate the Consistency:** To determine if the equation is consistent, we need to check if the system of linear equations represented by $Ax = p$ has at least one solution.3. **Solving the Equation:** If the equation is consistent, we can proceed to solve for $x$. - Use Gaussian Elimination or matrix row reductions to bring the matrix $A$ to its row echelon form. - Further reduce it to the reduced row echelon form (RREF) to easily identify solutions.4. **Expressing the Solution:** Once you have the RREF, you can express the solution in parametric vector form. This involves: - Identifying pivot columns and free variables. - Expressing the solution vector $x$ in terms of these free variables. 5. **Inconsistent Equation:** If the system is inconsistent (i.e., there is no combination of the variables that can satisfy the equation), state that the system has no solution.Summarizing:- **Solve $Ax = p$**: If consistent, find solution(s).- **Parametric Vector Form**: Represent solutions as vectors.- **Inconsistent Case**: Demonstrate the lack of solutions.In conclusion, solving $Ax = p$ involves matrix operations and understanding linear system consistency. This fundamental process helps in linear algebra applications and solving real-world problems.

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Answered: Solve the equation Ax = p, or show that…

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Solve the equation Ax = p, or show that the equation is inconsistent. Write your solution in parametric vector form.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.1: Matrix Operations

Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...

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Question

Write solution in parametric form from vector

### Understanding Matrix and Vector Operations

Consider the following matrix **A** and vectors **p** and **q**:

\[
A = \begin{bmatrix}
1 & -3 \\
-3 & 0 \\
5 & 2
\end{bmatrix}
\quad
p = \begin{bmatrix}
11 \\
-6 \\
4
\end{bmatrix}
\quad
q = \begin{bmatrix}
9 \\
-9 \\
11
\end{bmatrix}
\]

#### Matrix **A**
- **A** is a 3x2 matrix, meaning it has 3 rows and 2 columns.
- The entries of the matrix are as follows:
- First row: 1, -3
- Second row: -3, 0
- Third row: 5, 2

#### Vector **p**
- **p** is a 3x1 column vector.
- The entries of vector **p** are:
- First element: 11
- Second element: -6
- Third element: 4

#### Vector **q**
- **q** is also a 3x1 column vector.
- The entries of vector **q** are:
- First element: 9
- Second element: -9
- Third element: 11

These matrices and vectors can be used in a variety of linear algebra operations such as matrix multiplication, finding the dot product, and solving systems of linear equations. Understanding their structure is fundamental in linear algebra and various applications in mathematics, physics, and engineering.

**Solving Systems of Linear Equations**

In this section, we will tackle the problem presented below:

**Problem Statement:**

Solve the equation $Ax = p$, or show that the equation is inconsistent. Write your solution in parametric vector form.

**Explanation and Steps:**

1. **Identify the Given Equation:**
The given equation is $Ax = p$, where $A$ is a matrix, $x$ is the vector of variables, and $p$ is the resultant vector.

2. **Evaluate the Consistency:**
To determine if the equation is consistent, we need to check if the system of linear equations represented by $Ax = p$ has at least one solution.

3. **Solving the Equation:**
If the equation is consistent, we can proceed to solve for $x$.

- Use Gaussian Elimination or matrix row reductions to bring the matrix $A$ to its row echelon form.
- Further reduce it to the reduced row echelon form (RREF) to easily identify solutions.

4. **Expressing the Solution:**
Once you have the RREF, you can express the solution in parametric vector form. This involves:

- Identifying pivot columns and free variables.
- Expressing the solution vector $x$ in terms of these free variables.

5. **Inconsistent Equation:**
If the system is inconsistent (i.e., there is no combination of the variables that can satisfy the equation), state that the system has no solution.

Summarizing:
- **Solve $Ax = p$**: If consistent, find solution(s).
- **Parametric Vector Form**: Represent solutions as vectors.
- **Inconsistent Case**: Demonstrate the lack of solutions.

In conclusion, solving $Ax = p$ involves matrix operations and understanding linear system consistency. This fundamental process helps in linear algebra applications and solving real-world problems.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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