Size of the Augmented Matrix

Oct 13, 2025 by miodeya.com 128 views

Introduction

In this article, we will explore a linear system that represents the sales of large and small popcorn servings at a movie theater over two consecutive days. The system consists of two equations with two variables, and our goal is to find the size of the augmented matrix for this system.

The Linear System

The linear system is given by the following two equations:

42L + 816 = 383 59L + 735 = 8529

where L represents the number of large popcorn servings sold, and the other variables represent the number of small popcorn servings sold.

Augmented Matrix

An augmented matrix is a matrix that includes the coefficients of the variables in the linear system, as well as the constant terms. In this case, the augmented matrix for the linear system is:

| 42 | 816 | 383 | | 59 | 735 | 8529 |

Size of the Augmented Matrix

To find the size of the augmented matrix, we need to determine the number of rows and columns it has. In this case, the augmented matrix has 2 rows (one for each equation) and 3 columns (one for each variable and one for the constant term).

Calculating the Size of the Augmented Matrix

To calculate the size of the augmented matrix, we can use the following formula:

Size = Number of Rows x Number of Columns

In this case, the number of rows is 2, and the number of columns is 3, so the size of the augmented matrix is:

Size = 2 x 3 = 6

Conclusion

The Linear System

The linear system is given by the following two equations:

42L + 816 = 383 59L + 735 = 8529

where L represents the number of large popcorn servings sold, and the other variables represent the number of small popcorn servings sold.

Augmented Matrix

An augmented matrix is a matrix that includes the coefficients of the variables in the linear system, as well as the constant terms. In this case, the augmented matrix for the linear system is:

| 42 | 816 | 383 | | 59 | 735 | 8529 |

Calculating the Size of the Augmented Matrix

To calculate the size of the augmented matrix, we can use the following formula:

Size = Number of Rows x Number of Columns

In this case, the number of rows is 2, and the number of columns is 3, so the size of the augmented matrix is:

Size = 2 x 3 = 6

Properties of the Augmented Matrix

The augmented matrix has several properties that are important to note:

The number of rows in the augmented matrix is equal to the number of equations in the linear system.
The number of columns in the augmented matrix is equal to the number of variables in the linear system, plus one (for the constant term).
The augmented matrix can be used to solve the linear system using various methods, such as Gaussian elimination or LU decomposition.

Solving the Linear System

To solve the linear system, we can use various methods, such as Gaussian elimination or LU decomposition. These methods involve performing row operations on the augmented matrix to transform it into a form that is easier to solve.

Gaussian Elimination

Gaussian elimination is a method for solving linear systems that involves performing row operations on the augmented matrix to transform it into a form that is easier to solve. The steps involved in Gaussian elimination are:

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

| 1 | 0 | a13 | | 0 | 1 | a23 | | 0 | 0 | a33 |

The solution to the linear system is then given by:

L = a13 S = a23

LU Decomposition

LU decomposition is another method for solving linear systems that involves decomposing the augmented matrix into the product of two matrices, L and U. The steps involved in LU decomposition are:

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

| 1 | 0 | a13 | | 0 | 1 | a23 | | 0 | 0 | a33 |

The solution to the linear system is then given by:

L = a13 S = a23

Conclusion

In this article, we explored a linear system that represents the sales of large and small popcorn servings at a movie theater over two consecutive days. We found the size of the augmented matrix for this system, which is 6. We also discussed various methods for solving linear systems, including Gaussian elimination and LU decomposition.

The Linear System

The linear system is given by the following two equations:

42L + 816 = 383 59L + 735 = 8529

where L represents the number of large popcorn servings sold, and the other variables represent the number of small popcorn servings sold.

Augmented Matrix

An augmented matrix is a matrix that includes the coefficients of the variables in the linear system, as well as the constant terms. In this case, the augmented matrix for the linear system is:

| 42 | 816 | 383 | | 59 | 735 | 8529 |

Calculating the Size of the Augmented Matrix

To calculate the size of the augmented matrix, we can use the following formula:

Size = Number of Rows x Number of Columns

In this case, the number of rows is 2, and the number of columns is 3, so the size of the augmented matrix is:

Size = 2 x 3 = 6

Properties of the Augmented Matrix

The augmented matrix has several properties that are important to note:

The number of rows in the augmented matrix is equal to the number of equations in the linear system.
The number of columns in the augmented matrix is equal to the number of variables in the linear system, plus one (for the constant term).
The augmented matrix can be used to solve the linear system using various methods, such as Gaussian elimination or LU decomposition.

Solving the Linear System

Gaussian Elimination

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

| 1 | 0 | a13 | | 0 | 1 | a23 | | 0 | 0 | a33 |

The solution to the linear system is then given by:

L = a13 S = a23

LU Decomposition

LU decomposition is another method for solving linear systems that involves decomposing the augmented matrix into the product of two matrices, L and U. The steps involved in LU decomposition are:

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

Q&A: Frequently Asked Questions

Q: What is the linear system that represents the sales of large and small popcorn servings at a movie theater over two consecutive days? A: The linear system is given by the following two equations:

42L + 816 = 383 59L + 735 = 8529

where L represents the number of large popcorn servings sold, and the other variables represent the number of small popcorn servings sold.

Q: What is the size of the augmented matrix for this linear system? A: The size of the augmented matrix is 6, which is calculated by multiplying the number of rows (2) by the number of columns (3).

Q: What are the properties of the augmented matrix? A: The augmented matrix has several properties that are important to note:

The number of rows in the augmented matrix is equal to the number of equations in the linear system.
The number of columns in the augmented matrix is equal to the number of variables in the linear system, plus one (for the constant term).
The augmented matrix can be used to solve the linear system using various methods, such as Gaussian elimination or LU decomposition.

Q: How can we solve the linear system using Gaussian elimination? A: To solve the linear system using Gaussian elimination, we can perform row operations on the augmented matrix to transform it into a form that is easier to solve. The steps involved in Gaussian elimination are:

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

| 1 | 0 | a13 | | 0 | 1 | a23 | | 0 | 0 | a33 |

The solution to the linear system is then given by:

L = a13 S = a23

Q: How can we solve the linear system using LU decomposition? A: To solve the linear system using LU decomposition, we can decompose the augmented matrix into the product of two matrices, L and U. The steps involved in LU decomposition are:

Write the augmented matrix in the form:

| a11 | a12 | a13 | | a21 | a22 | a23 | | a31 | a32 | a33 |

Perform row operations to transform the matrix into the form:

| 1 | a12 | a13 | | 0 | a22 | a23 | | 0 | a32 | a33 |

Continue performing row operations until the matrix is in the form:

| 1 | 0 | a13 | | 0 | 1 | a23 | | 0 | 0 | a33 |

The solution to the linear system is then given by:

L = a13 S = a23

Q: What are the advantages of using Gaussian elimination to solve the linear system? A: The advantages of using Gaussian elimination to solve the linear system are:

It is a simple and straightforward method to use.
It can be used to solve systems with a large number of variables.
It can be used to solve systems with a large number of equations.

Q: What are the disadvantages of using Gaussian elimination to solve the linear system? A: The disadvantages of using Gaussian elimination to solve the linear system are:

It can be time-consuming to perform the row operations.
It can be difficult to perform the row operations if the matrix is large.
It can be difficult to determine the solution to the linear system if the matrix is singular.

Q: What are the advantages of using LU decomposition to solve the linear system? A: The advantages of using LU decomposition to solve the linear system are:

It is a more efficient method to use than Gaussian elimination.
It can be used to solve systems with a large number of variables.
It can be used to solve systems with a large number of equations.

Q: What are the disadvantages of using LU decomposition to solve the linear system? A: The disadvantages of using LU decomposition to solve the linear system are:

It can be difficult to perform the decomposition if the matrix is large.
It can be difficult to determine the solution to the linear system if the matrix is singular.
It can be difficult to perform the decomposition if the matrix is ill-conditioned.