Find An Equation Involving \[$g, H,\$\] And \[$k\$\] That Makes This Augmented Matrix Correspond To A Consistent System:$\[ \begin{bmatrix} 1 & -4 & 7 & G \\ 0 & 3 & -5 & H \\ -2 & 5 & -9 &

Introduction

In linear algebra, an augmented matrix is a matrix that includes an additional column to represent the constants on the right-hand side of a system of linear equations. The goal of this article is to find an equation involving the variables g, h, and k that makes the given augmented matrix correspond to a consistent system. A consistent system is one that has at least one solution.

Understanding the Augmented Matrix

The given augmented matrix is:

$\[ \begin{bmatrix} 1 & -4 & 7 & g \\ 0 & 3 & -5 & h \\ -2 & 5 & -9 & \end{bmatrix}$

To make this matrix correspond to a consistent system, we need to find values for g, h, and k that satisfy the system of linear equations represented by the matrix.

Row Operations

To analyze the matrix, we can perform row operations to transform it into row echelon form. The row echelon form of a matrix is a form where all the entries below the leading entry of each row are zeros.

Performing row operations on the given matrix, we get:

$\[ \begin{bmatrix} 1 & -4 & 7 & g \\ 0 & 3 & -5 & h \\ 0 & 13 & -13 & \end{bmatrix}$

Analyzing the Matrix

From the row echelon form of the matrix, we can see that the first two rows represent two linear equations:

1. $x_1 - 4x_2 + 7x_3 = g$
2. $3x_2 - 5x_3 = h$

The third row represents a linear combination of the first two rows:

$13x_2 - 13x_3 = 0$

Finding the Equation

To make the system consistent, we need to find values for g, h, and k that satisfy the system of linear equations. We can start by solving the second equation for $x_2$:

$x_2 = \frac{h + 5x_3}{3}$

Substituting this expression for $x_2$ into the first equation, we get:

$x_1 - 4\left(\frac{h + 5x_3}{3}\right) + 7x_3 = g$

Simplifying this equation, we get:

$x_1 - \frac{4h}{3} - \frac{20x_3}{3} + 7x_3 = g$

Combining like terms, we get:

$x_1 - \frac{4h}{3} + \frac{11x_3}{3} = g$

Conclusion

In conclusion, the equation involving g, h, and k that makes the given augmented matrix correspond to a consistent system is:

$x_1 - \frac{4h}{3} + \frac{11x_3}{3} = g$

This equation represents a linear relationship between the variables x1, x3, and the constants g and h.

Discussion

The equation we found is a specific example of a linear equation that makes the given augmented matrix correspond to a consistent system. In general, we can find an equation involving g, h, and k that makes the matrix correspond to a consistent system by performing row operations to transform the matrix into row echelon form and then analyzing the resulting matrix.

Example Use Case

Suppose we have a system of linear equations represented by the matrix:

$\[ \begin{bmatrix} 1 & -4 & 7 & 2 \\ 0 & 3 & -5 & 3 \\ -2 & 5 & -9 & \end{bmatrix}$

Using the equation we found, we can substitute the values of g and h into the equation to get:

$x_1 - \frac{4(3)}{3} + \frac{11x_3}{3} = 2$

Simplifying this equation, we get:

$x_1 - 4 + \frac{11x_3}{3} = 2$

Combining like terms, we get:

$x_1 + \frac{11x_3}{3} = 6$

This equation represents a linear relationship between the variables x1 and x3.

Final Thoughts

In this article, we found an equation involving g, h, and k that makes the given augmented matrix correspond to a consistent system. We also discussed how to use row operations to transform the matrix into row echelon form and analyze the resulting matrix. This equation can be used to solve systems of linear equations represented by the matrix.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon

Further Reading

• [1] "Row Operations and Row Echelon Form" by Math Is Fun
• [2] "Linear Equations and Matrices" by Khan Academy
  

Introduction

In our previous article, we found an equation involving g, h, and k that makes the given augmented matrix correspond to a consistent system. In this article, we will answer some frequently asked questions about the equation and the process of finding it.

Q: What is the purpose of finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system?

A: The purpose of finding such an equation is to solve systems of linear equations represented by the matrix. By finding an equation that makes the matrix correspond to a consistent system, we can determine the values of the variables that satisfy the system.

Q: How do you perform row operations to transform the matrix into row echelon form?

A: To perform row operations, you can use the following steps:

1. Swap two rows to get a leading entry in the desired position.
2. Multiply a row by a non-zero constant to get a leading entry of the desired value.
3. Add a multiple of one row to another row to get a leading entry in the desired position.

Q: What is the significance of the row echelon form of a matrix?

A: The row echelon form of a matrix is a form where all the entries below the leading entry of each row are zeros. This form makes it easier to analyze the matrix and determine the values of the variables that satisfy the system.

Q: How do you analyze the matrix in row echelon form?

A: To analyze the matrix in row echelon form, you can look at the leading entries of each row and determine the values of the variables that satisfy the system. You can also use the equations represented by the rows to solve for the variables.

Q: What is the equation involving g, h, and k that makes the given augmented matrix correspond to a consistent system?

A: The equation involving g, h, and k that makes the given augmented matrix correspond to a consistent system is:

$x_1 - \frac{4h}{3} + \frac{11x_3}{3} = g$

Q: How do you use the equation to solve systems of linear equations represented by the matrix?

A: To use the equation to solve systems of linear equations represented by the matrix, you can substitute the values of g and h into the equation and solve for the variables. You can also use the equation to determine the values of the variables that satisfy the system.

Q: What are some common mistakes to avoid when finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system?

A: Some common mistakes to avoid when finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system include:

• Not performing row operations correctly
• Not analyzing the matrix in row echelon form correctly
• Not using the equation correctly to solve systems of linear equations represented by the matrix

Q: How do you determine if a system of linear equations represented by a matrix is consistent or inconsistent?

A: To determine if a system of linear equations represented by a matrix is consistent or inconsistent, you can perform row operations to transform the matrix into row echelon form and analyze the resulting matrix. If the matrix is in row echelon form and all the entries below the leading entry of each row are zeros, then the system is consistent. If the matrix is not in row echelon form or there are non-zero entries below the leading entry of each row, then the system is inconsistent.

Q: What are some real-world applications of finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system?

A: Some real-world applications of finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system include:

• Solving systems of linear equations in physics and engineering
• Determining the values of variables in economics and finance
• Analyzing data in statistics and data science

Conclusion

In this article, we answered some frequently asked questions about finding an equation involving g, h, and k that makes the augmented matrix correspond to a consistent system. We also discussed some common mistakes to avoid and some real-world applications of the equation. By understanding the equation and the process of finding it, you can solve systems of linear equations represented by the matrix and determine the values of the variables that satisfy the system.