Perform each matrix row operation and write the new matrix.
$\begin{bmatrix}\)1 & -1 & 1 & 1 & $\vert$ & 3 \\0 & 1 & -2 & -1 & $\vert$ & 0 \\2 & 0 & 3 & 4 & $\vert$ & 11 \\5 & 1 & 2 & 4 & $\vert$ & 6$\end{bmatrix}\[\quad\]\begin{array}{l}$-2R_1 + R_3 \\-5R_1 + R_4\(\end{array}$

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Identify the given matrix and the row operations to perform. The matrix is: \[\left[\begin{array}{cccc|c} 1 & -1 & 1 & 1 & 3 \\ 0 & 1 & -2 & -1 & 0 \\ 2 & 0 & 3 & 4 & 11 \\ 5 & 2 & 4 & 6 & 6 \end{array}\right]\] and the row operations are: \[-2R_1 + R_3\] \[-5R_1 + R_4\]

Perform the first row operation: replace row 3 ($R_3$) with $-2$ times row 1 plus row 3. Calculate each element of the new row 3 by multiplying each element of row 1 by $-2$ and then adding the corresponding element of row 3.

Perform the second row operation: replace row 4 ($R_4$) with $-5$ times row 1 plus row 4. Calculate each element of the new row 4 by multiplying each element of row 1 by $-5$ and then adding the corresponding element of row 4.

Write the new matrix after these row operations, keeping rows 1 and 2 unchanged, and rows 3 and 4 updated with the results from the previous steps.

Double-check each element in the new rows 3 and 4 to ensure the arithmetic was done correctly, confirming the accuracy of the row operations.

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주요 개념

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Matrix Row Operations

Matrix row operations are fundamental techniques used to manipulate matrices to achieve a desired form. These include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. They are essential for solving systems of linear equations and finding matrix inverses.

Performing Linear Combinations of Rows

Performing linear combinations involves replacing a row by adding or subtracting a multiple of another row. For example, the operation -2R1 + R3 means multiplying row 1 by -2 and adding it to row 3. This helps simplify matrices and is key in methods like Gaussian elimination.

Augmented Matrices and Systems of Equations

An augmented matrix represents a system of linear equations, combining coefficients and constants in one matrix. Row operations on augmented matrices correspond to equivalent transformations of the system, preserving solutions while simplifying the matrix to solve for variables.