Give The Reduced Row Echelon Form Of The Augmented Matrix:$\[ \left[\begin{array}{cccc} -5 & 0 & 12 & 100 \\ 3 & 1 & 0 & 300 \\ 2 & 0 & -5 & 250 \end{array}\right] \\]

Introduction

In linear algebra, the reduced row echelon form (RREF) of a matrix is a fundamental concept used to solve systems of linear equations. The RREF is a unique matrix that can be obtained from any given matrix through a series of elementary row operations. In this article, we will focus on finding the RREF of the given augmented matrix.

What is an Augmented Matrix?

An augmented matrix is a matrix that combines the coefficients of a system of linear equations with the constant terms. It is denoted by the symbol $\left[\begin{array}{cccc}a & b & c & d\end{array}\right]$, where $a$, $b$, $c$, and $d$ are the coefficients and constant terms of the system of linear equations.

The Given Augmented Matrix

The given augmented matrix is:

$$\left[\begin{array}{cccc} -5 & 0 & 12 & 100 \\ 3 & 1 & 0 & 300 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 1: Interchange Rows

To begin the process of finding the RREF, we need to interchange rows to ensure that the row with the largest absolute value in the first column is at the top. In this case, we can interchange rows 1 and 2 to get:

$$\left[\begin{array}{cccc} 3 & 1 & 0 & 300 \\ -5 & 0 & 12 & 100 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 2: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 1 by $\frac{1}{3}$ to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ -5 & 0 & 12 & 100 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 3: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add 5 times row 1 to row 2 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 12 & 500 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 4: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 3 by $\frac{1}{2}$ to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 12 & 500 \\ 1 & 0 & -\frac{5}{2} & 125 \end{array}\right]$$

Step 5: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add -1 times row 3 to row 1 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 12 & 500 \\ 1 & 0 & -\frac{5}{2} & 125 \end{array}\right]$$

Step 6: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 3 by 2 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 12 & 500 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 7: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add -2 times row 3 to row 1 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 12 & 500 \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 8: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 2 by $\frac{1}{12}$ to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 9: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add -2 times row 2 to row 3 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 10: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 3 by $\frac{1}{2}$ to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 1 & 0 & -\frac{5}{2} & 125 \end{array}\right]$$

Step 11: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add -1 times row 3 to row 1 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 1 & 0 & -\frac{5}{2} & 125 \end{array}\right]$$

Step 12: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 3 by 2 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 13: Add a Multiple of a Row to Another Row

Now, we need to add a multiple of a row to another row to eliminate the entries below the first entry in the first column. In this case, we can add -2 times row 3 to row 1 to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 14: Multiply a Row by a Nonzero Constant

Next, we need to multiply a row by a nonzero constant to ensure that the first entry in the row is 1. In this case, we can multiply row 2 by $\frac{1}{12}$ to get:

$$\left[\begin{array}{cccc} 1 & \frac{1}{3} & 0 & 100 \\ 0 & 0 & 1 & \frac{500}{12} \\ 2 & 0 & -5 & 250 \end{array}\right]$$

Step 15: Add a Multiple of a Row to Another Row

Q&A: Reduced Row Echelon Form of Augmented Matrices

Q: What is the reduced row echelon form (RREF) of a matrix? A: The RREF of a matrix is a unique matrix that can be obtained from any given matrix through a series of elementary row operations. It is a fundamental concept used to solve systems of linear equations.

Q: What are the steps to find the RREF of an augmented matrix? A: The steps to find the RREF of an augmented matrix are:

1. Interchange rows to ensure that the row with the largest absolute value in the first column is at the top.
2. Multiply a row by a nonzero constant to ensure that the first entry in the row is 1.
3. Add a multiple of a row to another row to eliminate the entries below the first entry in the first column.
4. Repeat steps 2 and 3 until the matrix is in RREF.

Q: What is the purpose of finding the RREF of an augmented matrix? A: The purpose of finding the RREF of an augmented matrix is to solve systems of linear equations. The RREF provides a unique solution to the system of linear equations.

Q: How do I know when the matrix is in RREF? A: A matrix is in RREF when:

• All entries below the leading entry in each row are zero.
• The leading entry in each row is 1.
• The leading entry in each row is to the right of the leading entry in the row above it.

Q: What are some common mistakes to avoid when finding the RREF of an augmented matrix? A: Some common mistakes to avoid when finding the RREF of an augmented matrix are:

• Not following the correct order of operations.
• Not using the correct row operations.
• Not checking for leading entries in each row.

Q: Can I use a calculator or computer program to find the RREF of an augmented matrix? A: Yes, you can use a calculator or computer program to find the RREF of an augmented matrix. However, it is still important to understand the steps involved in finding the RREF and to be able to explain the process.

Q: How do I apply the RREF to real-world problems? A: The RREF can be applied to real-world problems in a variety of fields, including physics, engineering, and economics. For example, the RREF can be used to solve systems of linear equations that model real-world problems, such as the motion of objects or the flow of fluids.

Q: What are some common applications of the RREF? A: Some common applications of the RREF include:

• Solving systems of linear equations.
• Finding the inverse of a matrix.
• Determining the rank of a matrix.
• Solving systems of linear inequalities.

Q: Can I use the RREF to solve systems of linear inequalities? A: Yes, you can use the RREF to solve systems of linear inequalities. However, the RREF is typically used to solve systems of linear equations, and the process for solving systems of linear inequalities is slightly different.

Q: How do I know if a matrix is in RREF? A: A matrix is in RREF when:

• All entries below the leading entry in each row are zero.
• The leading entry in each row is 1.
• The leading entry in each row is to the right of the leading entry in the row above it.

Q: What are some common mistakes to avoid when using the RREF to solve systems of linear equations? A: Some common mistakes to avoid when using the RREF to solve systems of linear equations are:

• Not following the correct order of operations.
• Not using the correct row operations.
• Not checking for leading entries in each row.

Q: Can I use the RREF to solve systems of linear equations with more than three variables? A: Yes, you can use the RREF to solve systems of linear equations with more than three variables. However, the process may be more complex and may require the use of additional row operations.

Q: How do I apply the RREF to systems of linear equations with more than three variables? A: To apply the RREF to systems of linear equations with more than three variables, you can follow the same steps as before, but you may need to use additional row operations to eliminate the entries below the leading entry in each row.