Math Problem Statement

Perform the row operation(s) on the given augmented matrix with operations: (a) R2 = 3R1 + R2, (b) R3 = -2R1 + R3, (c) R3 = 4R2 + R3, and identify the correct resulting matrix from the options provided.

Solution

Let's go through the row operations provided step by step and apply them to the augmented matrix:

Initial Matrix

1 & 2 & -3 & | & 4 \\ -3 & -5 & 8 & | & -10 \\ 2 & 0 & -1 & | & 4 \\ \end{bmatrix}$$ ### Step-by-Step Row Operations 1. **Operation (a)**: $$R_2 = 3R_1 + R_2$$ \[ R_2 = 3 \cdot [1, 2, -3, |, 4] + [-3, -5, 8, |, -10] \] Calculating each element: - First element: $$3 \cdot 1 + (-3) = 0$$ - Second element: $$3 \cdot 2 + (-5) = 1$$ - Third element: $$3 \cdot (-3) + 8 = -1$$ - Fourth element (augmented part): $$3 \cdot 4 + (-10) = 2$$ After this operation, the matrix becomes: $$\begin{bmatrix} 1 & 2 & -3 & | & 4 \\ 0 & 1 & -1 & | & 2 \\ 2 & 0 & -1 & | & 4 \\ \end{bmatrix}$$ 2. **Operation (b)**: $$R_3 = -2R_1 + R_3$$ \[ R_3 = -2 \cdot [1, 2, -3, |, 4] + [2, 0, -1, |, 4] \] Calculating each element: - First element: $$-2 \cdot 1 + 2 = 0$$ - Second element: $$-2 \cdot 2 + 0 = -4$$ - Third element: $$-2 \cdot (-3) + (-1) = 5$$ - Fourth element (augmented part): $$-2 \cdot 4 + 4 = -4$$ After this operation, the matrix becomes: $$\begin{bmatrix} 1 & 2 & -3 & | & 4 \\ 0 & 1 & -1 & | & 2 \\ 0 & -4 & 5 & | & -4 \\ \end{bmatrix}$$ 3. **Operation (c)**: $$R_3 = 4R_2 + R_3$$ \[ R_3 = 4 \cdot [0, 1, -1, |, 2] + [0, -4, 5, |, -4] \] Calculating each element: - First element: $$4 \cdot 0 + 0 = 0$$ - Second element: $$4 \cdot 1 + (-4) = 0$$ - Third element: $$4 \cdot (-1) + 5 = 1$$ - Fourth element (augmented part): $$4 \cdot 2 + (-4) = 4$$ After this operation, the matrix becomes: $$\begin{bmatrix} 1 & 2 & -3 & | & 4 \\ 0 & 1 & -1 & | & 2 \\ 0 & 0 & 1 & | & 4 \\ \end{bmatrix}$$ ### Final Answer The matrix matches **Option A**: $$\begin{bmatrix} 1 & 2 & -3 & | & 4 \\ 0 & 1 & -1 & | & 2 \\ 0 & 0 & 1 & | & 4 \\ \end{bmatrix}$$ --- Would you like more details on the row operations? ### Related Questions 1. What are augmented matrices and how are they used in solving linear systems? 2. How do elementary row operations help in transforming matrices? 3. What are the different types of row operations in matrix algebra? 4. How can we identify the row echelon form of a matrix? 5. What does the final form of this matrix indicate about the solution of the linear system? ### Tip To simplify row operations, focus on creating leading ones and zeroes in strategic locations to achieve row echelon or reduced row echelon form.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Row Operations
Augmented Matrix

Formulas

Row operation formulas for matrix transformation

Theorems

Matrix row operations to achieve row echelon form

Suitable Grade Level

Grades 10-12

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