Perform the elementary row operation $-\frac{1}{3} R_2 \rightarrow R_2$ on the given matrix. Write numbers as integers or simplified fractions.
$\left[\begin{array}{ccc:c}
3 & 5 & 1 & 4 \\
1 & 3 & 1 & 5 \\
1 & 3 & 0 & 10
\end{array}\right]$
Resulting matrix:
Answer (2)
Multiply the second row of the matrix by − 3 1 .
The new second row is [ − 3 1 , − 1 , − 3 1 , − 3 5 ] .
Replace the original second row with the new row.
The resulting matrix is 3 − 3 1 1 5 − 1 3 1 − 3 1 0 4 − 3 5 10 .
Explanation
Understanding the Problem We are given a 3x4 matrix and asked to perform the elementary row operation − 3 1 R 2 → R 2 . This means we need to multiply each element in the second row of the matrix by − 3 1 . The first and third rows will remain unchanged.
Applying the Row Operation The given matrix is: 3 1 1 5 3 3 1 1 0 4 5 10 We need to multiply the second row by − 3 1 .
Calculating the New Row Multiplying the second row by − 3 1 , we get: − 3 1 × [ 1 , 3 , 1 , 5 ] = [ − 3 1 , − 1 , − 3 1 , − 3 5 ]
Constructing the Resulting Matrix Now, we replace the second row of the original matrix with this new row. The resulting matrix is: 3 − 3 1 1 5 − 1 3 1 − 3 1 0 4 − 3 5 10
Final Answer Therefore, the resulting matrix after performing the row operation is: 3 − 3 1 1 5 − 1 3 1 − 3 1 0 4 − 3 5 10
Examples
Elementary row operations are fundamental in solving systems of linear equations, which arise in various fields such as engineering, economics, and computer science. For instance, in structural analysis, these operations can help determine the forces acting on different parts of a bridge. In economics, they can be used to find equilibrium prices in a market. In computer graphics, they are essential for performing transformations on objects in 3D space. By manipulating matrices using row operations, we can simplify complex problems and find solutions more efficiently.
By performing the row operation − 3 1 R 2 → R 2 , we multiply all elements of the second row of the matrix by − 3 1 . The resulting matrix is 3 − 3 1 1 5 − 1 3 1 − 3 1 0 4 − 3 5 10 .
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