Math Problem Statement
Perform the indicated row operations on the given augmented matrix.
6 -3 1 1
-5 -2 5 -4
5 6 4 2
a.) Interchange row 2 and row 1
Correct
b.) Multiply row 2 by 3 and add to row 3
c.) Multiply row 1 by 3
Solution
Let’s perform the indicated row operations step by step on the given augmented matrix. The matrix is:
6 & -3 & 1 & 1 \\ -5 & -2 & 5 & -4 \\ 5 & 6 & 4 & 2 \end{bmatrix}$$ ### a.) Interchange row 2 and row 1: After interchanging row 2 and row 1, the matrix becomes: $$\begin{bmatrix} -5 & -2 & 5 & -4 \\ 6 & -3 & 1 & 1 \\ 5 & 6 & 4 & 2 \end{bmatrix}$$ --- ### b.) Multiply row 2 by 3 and add to row 3: 1. Start with the matrix from part (a): $$\begin{bmatrix} -5 & -2 & 5 & -4 \\ 6 & -3 & 1 & 1 \\ 5 & 6 & 4 & 2 \end{bmatrix}$$ 2. Multiply row 2 by $$3$$: $$3R_2 = [18, -9, 3, 3]$$ 3. Add $$3R_2$$ to row 3: \[ R_3 = R_3 + 3R_2 = [5, 6, 4, 2] + [18, -9, 3, 3] = [23, -3, 7, 5] \] The matrix now becomes: $$\begin{bmatrix} -5 & -2 & 5 & -4 \\ 6 & -3 & 1 & 1 \\ 23 & -3 & 7 & 5 \end{bmatrix}$$ --- ### c.) Multiply row 1 by 3: 1. Start with the matrix from part (b): $$\begin{bmatrix} -5 & -2 & 5 & -4 \\ 6 & -3 & 1 & 1 \\ 23 & -3 & 7 & 5 \end{bmatrix}$$ 2. Multiply row 1 by $$3$$: $$3R_1 = [-15, -6, 15, -12]$$ The matrix now becomes: $$\begin{bmatrix} -15 & -6 & 15 & -12 \\ 6 & -3 & 1 & 1 \\ 23 & -3 & 7 & 5 \end{bmatrix}$$ --- ### Final Result: After all operations, the final matrix is: $$\begin{bmatrix} -15 & -6 & 15 & -12 \\ 6 & -3 & 1 & 1 \\ 23 & -3 & 7 & 5 \end{bmatrix}$$ Let me know if you need further clarification or steps! --- ### 5 Related Questions: 1. How do you determine when to perform row operations in solving systems of equations? 2. Can this matrix be reduced to row-echelon form? If so, what are the next steps? 3. How does interchanging rows affect the determinant of the matrix? 4. What is the significance of augmented matrices in solving linear systems? 5. Can this system of equations be solved using inverse matrices? --- ### Tip: Always label your steps clearly when performing row operations to avoid losing track of changes in the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Row Operations
Augmented Matrices
Formulas
Interchange two rows: Swap the positions of the two rows.
Multiply a row by a scalar: Multiply each element of the row by the scalar value.
Add a multiple of one row to another: New row = Current row + Scalar × Another row.
Theorems
Elementary Row Operations
Suitable Grade Level
Grades 10-12 or Early College
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