Verified Answer :
- Operation 45: \(R_3 + R_2 \to R_3\).
- Operation 46: \(R_3 - 2R_1 \to R_3\).
- Operation 47: \(R_4 + 3R_1 \to R_4\).
- Operation 48: \(R_2 - R_1 \to R_2\).
- These operations represent adding a multiple of one row to another, a fundamental step in Gaussian elimination. \(\boxed{R_3+R_2, R_3-2R_1, R_4+3R_1, R_2-R_1}\)
Explanation
- Problem Analysis
Elementary row operations allow us to manipulate matrices to simplify them or to find their reduced row echelon forms. The three standard operations are: swapping two rows (\(R_i \leftrightarrow R_j\)), multiplying a row by a non-zero scalar (\(kR_i \to R_i\)), and adding a multiple of one row to another row (\(R_i + kR_j \to R_i\)). We analyze each given pair to determine which transformation was applied to the initial matrix to reach the final matrix.
- Matrix Operation 45
For the first matrix pair:
\[\left[\begin{array}{rrr:r}1 & 5 & -6 & 3 \ 0 & 1 & -3 & -2 \ 0 & -1 & 2 & 1\end{array}\right] \rightarrow\left[\begin{array}{lll:r}1 & 5 & -6 & 3 \ 0 & 1 & -3 & -2 \ 0 & 0 & -1 & -1\end{array}\right]\]
Comparing the rows, rows 1 and 2 remain unchanged. Row 3 changes from \((0, -1, 2, 1)\) to \((0, 0, -1, -1)\). We test the operation \(R_3 + kR_2 \to R_3\). Using the second column, we have \(-1 + k(1) = 0\), which implies \(k = 1\). Verifying this with other elements: \(2 + 1(-3) = -1\) and \(1 + 1(-2) = -1\). Thus, the operation performed is \(R_3 + R_2 \to R_3\).
- Matrix Operation 46
For the second matrix pair:
\[\left[\begin{array}{rrr:r}3 & 1 & -5 & 4 \ 9 & -1 & 4 & -2 \ 8 & 4 & -3 & 1\end{array}\right] \rightarrow\left[\begin{array}{rrr:r}3 & 1 & -5 & 4 \ 9 & -1 & 4 & -2 \ 2 & 2 & 7 & -7\end{array}\right]\]
Rows 1 and 2 remain unchanged. Row 3 changes from \((8, 4, -3, 1)\) to \((2, 2, 7, -7)\). We test \(R_3 + kR_1 \to R_3\). Using the first column, \(8 + k(3) = 2 \Rightarrow 3k = -6 \Rightarrow k = -2\). Verifying: \(4 + (-2)(1) = 2\), \(-3 + (-2)(-5) = 7\), and \(1 + (-2)(4) = -7\). Thus, the operation performed is \(R_3 - 2R_1 \to R_3\).
- Matrix Operation 47
For the third matrix pair:
\[\left[\begin{array}{rrrr:r}1 & 15 & 2 & 4 & 14 \ 0 & 8 & 5 & -5 & 15 \ 2 & 1 & 0 & 16 & 20 \ -3 & -11 & -1 & 6 & -4\end{array}\right] \rightarrow\left[\begin{array}{rrrr:r}1 & 15 & 2 & 4 & 14 \ 0 & 8 & 5 & -5 & 15 \ 2 & 1 & 0 & 16 & 20 \ 0 & 34 & 5 & 18 & 38\end{array}\right]\]
Rows 1, 2, and 3 remain unchanged. Row 4 changes from \((-3, -11, -1, 6, -4)\) to \((0, 34, 5, 18, 38)\). Testing \(R_4 + kR_1 \to R_4\), we use the first column: \(-3 + k(1) = 0 \Rightarrow k = 3\). Verifying: \(-11 + 3(15) = 34\), \(-1 + 3(2) = 5\), \(6 + 3(4) = 18\), and \(-4 + 3(14) = 38\). Thus, the operation performed is \(R_4 + 3R_1 \to R_4\).
- Matrix Operation 48
For the fourth matrix pair:
\[\left[\begin{array}{rrrrr:r}8 & -2 & 0 & 2 & 12 & -2 \ 8 & 5 & -7 & 1 & 6 & 9 \ -1 & 0 & 9 & 3 & 3 & 2\end{array}\right] \rightarrow\left[\begin{array}{rrrrr:r}8 & -2 & 0 & 2 & 12 & -2 \ 0 & 7 & -7 & -1 & -6 & 11 \ -1 & 0 & 9 & 3 & 3 & 2\end{array}\right]\]
Rows 1 and 3 remain unchanged. Row 2 changes from \((8, 5, -7, 1, 6, 9)\) to \((0, 7, -7, -1, -6, 11)\). Testing \(R_2 + kR_1 \to R_2\), we use the first column: \(8 + k(8) = 0 \Rightarrow k = -1\). Verifying: \(5 + (-1)(-2) = 7\), \(-7 + (-1)(0) = -7\), \(1 + (-1)(2) = -1\), \(6 + (-1)(12) = -6\), and \(9 + (-1)(-2) = 11\). Thus, the operation performed is \(R_2 - R_1 \to R_2\).
- Conclusion
In summary, the row operations for the matrices are:
- \(R_3 + R_2 \to R_3\)
- \(R_3 - 2R_1 \to R_3\)
- \(R_4 + 3R_1 \to R_4\)
- \(R_2 - R_1 \to R_2\)
Examples
In data science and statistics, identifying patterns in datasets is crucial. Just as you performed row operations to transform these matrices, you might need to adjust features in a dataset to make them more manageable or interpretable. For example, if you are standardizing scores by subtracting a mean or scaling data by a factor, you are effectively performing the same types of linear row transformations that help reveal insights into the underlying variables. Mastering these matrix operations forms the basis for algorithms like Principal Component Analysis or linear regression, which are widely used to simplify complex models.