Step-by-Step Guidance: Row Reduction to RREF for Systems of Linear Equations LO9

Q1. Find the RREF of the augmented matrix for the following system of equations:

$2x + 5y + z = 1$ $x + 2y - z = -3$ $-x + 4y - 2z = 3$

Background

Topic: Row Reduction and RREF (Reduced Row Echelon Form)

This question tests your ability to convert a system of linear equations into an augmented matrix and use row operations to reduce it to RREF, which is a key skill in solving systems of equations.

Key Terms and Formulas

• Augmented Matrix: A matrix that represents a system of equations, including the constants on the right side of the equations.

• Row Operations: Operations you can perform on the rows of a matrix: swapping rows, multiplying a row by a nonzero constant, and adding/subtracting multiples of one row to another.

• RREF (Reduced Row Echelon Form): A matrix form where each leading entry is 1, each leading 1 is the only nonzero entry in its column, and each leading 1 is to the right of any leading 1s in the rows above.

Step-by-Step Guidance

1. Write the augmented matrix for the system:

   $\begin{bmatrix} 2 & 5 & 1 & | & 1 \\ 1 & 2 & -1 & | & -3 \\ -1 & 4 & -2 & | & 3 \end{bmatrix}$

2. Start by making the entry in the first row, first column a 1. You can do this by swapping rows or dividing the first row by 2, or by using the second row (which already has a 1 in the first column) as your starting point. Consider swapping Row 1 and Row 2 for easier calculations.

3. Use row operations to create zeros below the leading 1 in the first column. For example, subtract appropriate multiples of the new Row 1 from Rows 2 and 3 to eliminate the $x$-terms in those rows.

4. Continue the process for the second column: make the leading entry in the second row a 1 (if it isn't already), and use row operations to create zeros above and below this leading 1.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & -1 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$

This is the RREF of the augmented matrix, showing the unique solution for $x$, $y$, and $z$.

Q2. Find the RREF of the augmented matrix for the following system of equations:

$2x + 7y + z = 13$ $x + 3y - z = 9$ $-x + y - z = -1$

Background

Topic: Row Reduction and RREF

This question also tests your ability to use row operations to reduce a 3x3 system to RREF.

Key Terms and Formulas

• Same as above: Augmented Matrix, Row Operations, RREF.

Step-by-Step Guidance

1. Write the augmented matrix for the system:

   $\begin{bmatrix} 2 & 7 & 1 & | & 13 \\ 1 & 3 & -1 & | & 9 \\ -1 & 1 & -1 & | & -1 \end{bmatrix}$

2. Make the first entry in the first column a 1. You can do this by swapping rows or dividing the first row by 2, or by using the second row as the new first row.

3. Use row operations to create zeros below and above the leading 1 in the first column.

4. Proceed to the second column: make the leading entry in the second row a 1, and use row operations to create zeros above and below it.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 1 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$

This is the RREF of the augmented matrix, showing the unique solution for $x$, $y$, and $z$.

Q3. Find the RREF of the augmented matrix for the following system of equations:

$2x + 7y + 6z = 4$ $x + 4y + 5z = 3$ $-x - 2y + 3z = -1$

Background

Topic: Row Reduction and RREF

This question continues to test your ability to perform row operations to reach RREF for a 3x3 system.

Key Terms and Formulas

• Same as above: Augmented Matrix, Row Operations, RREF.

Step-by-Step Guidance

1. Write the augmented matrix for the system:

   $\begin{bmatrix} 2 & 7 & 6 & | & 4 \\ 1 & 4 & 5 & | & 3 \\ -1 & -2 & 3 & | & -1 \end{bmatrix}$

2. Make the first entry in the first column a 1. You can do this by swapping rows or dividing the first row by 2, or by using the second row as the new first row.

3. Use row operations to create zeros below and above the leading 1 in the first column.

4. Proceed to the second column: make the leading entry in the second row a 1, and use row operations to create zeros above and below it.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 1 \\ 0 & 0 & 1 & | & -1 \end{bmatrix}$

This is the RREF of the augmented matrix, showing the unique solution for $x$, $y$, and $z$.

Q4. Find the RREF of the augmented matrix for the following system of equations:

$3x + y + 2z = 16$ $2x + 2y + 3z = 16$ $2x - y + z = 11$

Background

Topic: Row Reduction and RREF

This question tests your ability to perform row operations to reach RREF for a 3x3 system.

Key Terms and Formulas

• Same as above: Augmented Matrix, Row Operations, RREF.

Step-by-Step Guidance

1. Write the augmented matrix for the system:

   $\begin{bmatrix} 3 & 1 & 2 & | & 16 \\ 2 & 2 & 3 & | & 16 \\ 2 & -1 & 1 & | & 11 \end{bmatrix}$

2. Make the first entry in the first column a 1. You can do this by dividing the first row by 3 or by swapping with another row if it makes calculations easier.

3. Use row operations to create zeros below and above the leading 1 in the first column.

4. Proceed to the second column: make the leading entry in the second row a 1, and use row operations to create zeros above and below it.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & 3 \\ 0 & 0 & 1 & | & 4 \end{bmatrix}$

This is the RREF of the augmented matrix, showing the unique solution for $x$, $y$, and $z$.

Q5. Find the RREF of the augmented matrix for the following system of equations:

$x + 3y + 4z = 11$ $3x + 15y + 10z = 15$ $2x + 11y + 5z = 3$

Background

Topic: Row Reduction and RREF

This question tests your ability to perform row operations to reach RREF for a 3x3 system.

Key Terms and Formulas

• Same as above: Augmented Matrix, Row Operations, RREF.

Step-by-Step Guidance

1. Write the augmented matrix for the system:

   $\begin{bmatrix} 1 & 3 & 4 & | & 11 \\ 3 & 15 & 10 & | & 15 \\ 2 & 11 & 5 & | & 3 \end{bmatrix}$

2. Make the first entry in the first column a 1 (already done in this case).

3. Use row operations to create zeros below the leading 1 in the first column by subtracting appropriate multiples of Row 1 from Rows 2 and 3.

4. Proceed to the second column: make the leading entry in the second row a 1, and use row operations to create zeros above and below it.

Try solving on your own before revealing the answer!

Final Answer:

$\begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & 3 \end{bmatrix}$

This is the RREF of the augmented matrix, showing the unique solution for $x$, $y$, and $z$.