Q1(a). Given the augmented matrix:

Background

Topic: Solutions to Systems of Linear Equations (Row-Reduced Echelon Form)

This question tests your ability to interpret an augmented matrix in (R)REF form, determine the number of solutions, and describe all solutions if they exist.

Key Terms and Concepts:

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

• Row-Reduced Echelon Form (RREF): A matrix form that makes it easy to identify solutions to the system.

• Pivot: The first nonzero entry in a row (usually 1 in RREF).

• Free Variable: A variable that does not correspond to a pivot column and can take any value.

Step-by-Step Guidance

1. Interpret the matrix as a system of equations. Each row corresponds to an equation in variables (since there are 4 columns before the augmented column).

2. Identify the pivot columns (columns with leading 1s in each row) and determine which variables are basic and which are free.

3. Check for any row that would indicate an inconsistency (e.g., a row of all zeros except for the last entry).

4. Express the basic variables in terms of the free variables, if any.

Try solving on your own before revealing the answer!

Q1(b). Given the augmented matrix:

Background

Topic: Solutions to Systems of Linear Equations (Row-Reduced Echelon Form)

This question tests your ability to interpret an augmented matrix in RREF, determine the number of solutions, and write out all solutions if they exist.

Key Terms and Concepts:

• Unique Solution: Occurs when every variable corresponds to a pivot column and there are no inconsistent rows.

• Infinitely Many Solutions: Occurs when there are free variables (columns without pivots).

• No Solution: Occurs if there is a row with all zeros except for the last entry (augmented column).

Step-by-Step Guidance

1. Write out the system of equations represented by the matrix.

2. Identify the pivot positions and determine if each variable is basic or free.

3. Check for any inconsistent rows.

4. If all variables are basic, set up the equations to solve for each variable in order.

Try solving on your own before revealing the answer!

Q1(c). Given the augmented matrix:

Background

Topic: Solutions to Systems of Linear Equations (Row-Reduced Echelon Form)

This question tests your ability to interpret an augmented matrix in RREF, determine the number of solutions, and write out all solutions if they exist.

Key Terms and Concepts:

• Back Substitution: Solving for variables starting from the bottom row upwards.

• Pivot Columns: Columns with leading 1s in each row.

Step-by-Step Guidance

1. Translate each row into an equation in (or as many variables as there are columns before the augmented column).

2. Identify which variables are basic and which are free.

3. Check for any inconsistencies in the system.

4. Use back substitution to express the solution(s).

Try solving on your own before revealing the answer!

Q1(d). Given the augmented matrix:

Background

Topic: Solutions to Systems of Linear Equations (Row-Reduced Echelon Form)

This question tests your ability to interpret an augmented matrix in RREF, determine the number of solutions, and write out all solutions if they exist.

Key Terms and Concepts:

• Free Variables: Variables that can take any value, leading to infinitely many solutions.

• Consistent System: No row leads to a contradiction.

Step-by-Step Guidance

1. Write out the equations represented by each row.

2. Identify which variables are basic (correspond to pivots) and which are free.

3. Express the basic variables in terms of the free variables.

4. Check for consistency (no row of the form , ).

Try solving on your own before revealing the answer!

Q1(e). Given the augmented matrix:

Background

Topic: Solutions to Systems of Linear Equations (Row-Reduced Echelon Form)

This question tests your ability to recognize when a system has a unique solution and to write out that solution.

Key Terms and Concepts:

• Unique Solution: Each variable corresponds to a pivot column and there are no inconsistencies.

Step-by-Step Guidance

1. Write out the equations for each variable directly from the matrix.

2. Check that each variable is basic (no free variables).

3. Verify that there are no inconsistent rows.

4. Set up the solution for each variable as indicated by the matrix.

Try solving on your own before revealing the answer!