Math Problem Statement

Wilma performs the elementary row operation represented by 2R_1 - R_3 on matrix A.

Solution

The matrix AA is given as:

-1 & 2 \\ 0 & 4 \\ 6 & 3 \\ -3 & -4 \end{bmatrix}$$ The row operation to be performed is: $$2R_1 - R_3$$ This means we take twice the first row and subtract the third row from it, replacing the third row with the result. Let’s perform the operation step by step. ### Step 1: Multiply the first row by 2 $$2R_1 = 2 \times \begin{bmatrix} -1 & 2 \end{bmatrix} = \begin{bmatrix} -2 & 4 \end{bmatrix}$$ ### Step 2: Subtract the third row from this result $$2R_1 - R_3 = \begin{bmatrix} -2 & 4 \end{bmatrix} - \begin{bmatrix} 6 & 3 \end{bmatrix} = \begin{bmatrix} -2 - 6 & 4 - 3 \end{bmatrix} = \begin{bmatrix} -8 & 1 \end{bmatrix}$$ So, the new third row will be $$\begin{bmatrix} -8 & 1 \end{bmatrix}$$. ### Step 3: The resulting matrix After performing this operation, the matrix becomes: $$\begin{bmatrix} -1 & 2 \\ 0 & 4 \\ -8 & 1 \\ -3 & -4 \end{bmatrix}$$ Let me know if you'd like a breakdown of the steps or further clarifications! ### Related Questions: 1. How do different elementary row operations affect matrix properties like determinant? 2. What is the importance of row operations in solving linear equations? 3. How can row operations be used to find the inverse of a matrix? 4. What other transformations could be applied to matrix $$A$$ to achieve row echelon form? 5. How does performing row operations affect the rank of a matrix? **Tip:** Row operations are particularly useful in Gaussian elimination to reduce a matrix to its row echelon or reduced row echelon form.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Elementary Row Operations

Formulas

2R_1 - R_3: Multiply the first row by 2, then subtract the third row from this result

Theorems

Elementary Row Operations: These operations are used to manipulate matrices and are fundamental in solving systems of linear equations and finding inverses.

Suitable Grade Level

High School (Grades 9-12)