Solving For U Using Cramer's Rule A Step-by-Step Guide With Row Reduction

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In the realm of linear algebra, Cramer's Rule stands as a powerful technique for solving systems of linear equations. It provides a direct method to find the value of each variable, bypassing the need for substitution or elimination. In this comprehensive guide, we'll tackle the following system of equations, focusing on isolating the variable 'u' using Cramer's Rule:

2u + x + y + z = 3
-8u - 7x - 3y + 5z = -3
u + 4x + y + z = 6
u + 3x + 7y - z = 1

We will embark on a journey to first evaluate the necessary determinants using row reduction, a pivotal technique for simplifying these calculations. This step-by-step approach will not only unveil the solution for 'u' but also illuminate the intricacies of Cramer's Rule and determinant evaluation. Let's delve into the world of linear systems and uncover the value of 'u'.

Cramer's Rule: A Powerful Tool for Solving Linear Systems

Before diving into the specifics of our problem, let's take a moment to appreciate the elegance and utility of Cramer's Rule. At its core, this rule offers a formulaic approach to solving systems of linear equations, where the number of equations matches the number of unknowns. Its beauty lies in its directness: by calculating determinants, we can isolate the value of each variable without the iterative processes often required by other methods.

Imagine a system of 'n' linear equations with 'n' unknowns, represented in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector. Cramer's Rule dictates that the solution for each variable can be expressed as a ratio of determinants. The denominator is the determinant of the coefficient matrix A, while the numerator is the determinant of a modified matrix formed by replacing the column corresponding to the variable of interest with the constant vector b.

However, the effectiveness of Cramer's Rule hinges on the invertibility of the coefficient matrix A. If the determinant of A is zero, the system either has no solution or infinitely many solutions, rendering Cramer's Rule inapplicable. In such cases, alternative methods like Gaussian elimination or matrix inversion are needed to analyze the system's behavior.

For systems where the determinant of A is non-zero, Cramer's Rule provides a clean and efficient way to obtain solutions. It's particularly advantageous when we need to find the value of only one or a few variables, as it avoids solving for the entire set of unknowns. However, for large systems, the computational cost of calculating multiple determinants can become significant, making other methods more practical.

In the context of our problem, we'll see how Cramer's Rule allows us to isolate 'u' by calculating determinants. But first, we'll master the art of row reduction, a technique that will make our determinant calculations much more manageable.

The Power of Row Reduction in Determinant Evaluation

Determinants, the cornerstone of Cramer's Rule, can be computationally intensive to calculate, especially for larger matrices. This is where row reduction comes into play. Row reduction, also known as Gaussian elimination, is a systematic process of transforming a matrix into an equivalent form that makes determinant calculation significantly easier. The beauty of row reduction lies in its ability to manipulate a matrix without altering its determinant (except for a possible sign change), paving the way for efficient computation.

The key to row reduction is the application of elementary row operations. These operations, when applied strategically, can transform a matrix into an upper triangular form, a matrix where all entries below the main diagonal are zero. The determinant of an upper triangular matrix is simply the product of its diagonal entries, a calculation far simpler than direct determinant expansion.

The three elementary row operations are:

  1. Swapping two rows: This operation changes the sign of the determinant.
  2. Multiplying a row by a non-zero constant: This operation multiplies the determinant by the same constant.
  3. Adding a multiple of one row to another: This operation leaves the determinant unchanged.

By carefully applying these operations, we can systematically eliminate entries below the main diagonal, transforming the matrix into an upper triangular form. The number of row swaps we perform is crucial, as each swap changes the sign of the determinant. By keeping track of these swaps and the constants we multiply rows by, we can accurately determine the determinant of the original matrix from the determinant of its row-reduced form.

In our quest to solve for 'u', we'll leverage row reduction to simplify the determinant calculations required by Cramer's Rule. This approach will not only make the computations feasible but also provide a deeper understanding of how matrix manipulation can streamline problem-solving in linear algebra.

Step-by-Step Application to Our System of Equations

Let's now apply these concepts to our given system of equations:

2u + x + y + z = 3
-8u - 7x - 3y + 5z = -3
u + 4x + y + z = 6
u + 3x + 7y - z = 1

1. Constructing the Coefficient Matrix and Constant Vector

First, we express the system in matrix form, identifying the coefficient matrix 'A' and the constant vector 'b':

A = | 2  1  1  1 |
    |-8 -7 -3  5 |
    | 1  4  1  1 |
    | 1  3  7 -1 |

b = | 3 |
    |-3 |
    | 6 |
    | 1 |

2. The Determinant of the Coefficient Matrix (det(A))

To use Cramer's Rule, we need det(A), also called delta. We'll employ row reduction to simplify the calculation. Let's start by swapping Row 1 and Row 3:

A' = | 1  4  1  1 |
     |-8 -7 -3  5 |
     | 2  1  1  1 |
     | 1  3  7 -1 |

This swap changes the sign of the determinant, so we'll keep track of this. Now, we'll perform row operations to eliminate the entries below the leading 1 in the first column:

  • Add 8 times Row 1 to Row 2 (R2 = R2 + 8R1)
  • Subtract 2 times Row 1 from Row 3 (R3 = R3 - 2R1)
  • Subtract Row 1 from Row 4 (R4 = R4 - R1)

The matrix becomes:

A' = | 1  4  1  1 |
     | 0 25  5 13 |
     | 0 -7 -1 -1 |
     | 0 -1  6 -2 |

Now, focus on the second column. Let's swap Row 2 and Row 4:

A' = | 1  4  1  1 |
     | 0 -1  6 -2 |
     | 0 -7 -1 -1 |
     | 0 25  5 13 |

Another sign change! Next, eliminate entries below the -1 in the second column:

  • Subtract 7 times Row 2 from Row 3 (R3 = R3 - 7R2)
  • Add 25 times Row 2 to Row 4 (R4 = R4 + 25R2)
A' = | 1  4   1   1 |
     | 0 -1   6  -2 |
     | 0  0 -43  13 |
     | 0  0 155 -37 |

Finally, let's eliminate the entry below -43 in the third column. We'll add 155/43 times Row 3 to Row 4 (R4 = R4 + (155/43)R3):

A' = |  1   4    1     1       |
     |  0  -1    6    -2       |
     |  0   0  -43    13       |
     |  0   0   0  -24/43   |

The determinant of this upper triangular matrix is the product of the diagonal entries: (1) * (-1) * (-43) * (-24/43) = -24. However, we performed two row swaps, each changing the sign of the determinant. Therefore, the determinant of the original matrix A is:

det(A) = (-1) * (-1) * (-24) = -24

3. Constructing the Matrix for 'u' (Au)

To find 'u', we replace the first column of A (the 'u' column) with the constant vector 'b':

Au = |  3   1   1   1 |
     | -3  -7  -3   5 |
     |  6   4   1   1 |
     |  1   3   7  -1 |

4. Determinant of Au (det(Au))

Again, we use row reduction. Swap Row 1 and Row 4:

Au' = |  1   3   7  -1 |
      | -3  -7  -3   5 |
      |  6   4   1   1 |
      |  3   1   1   1 |

Sign change! Now, eliminate entries below the leading 1 in the first column:

  • Add 3 times Row 1 to Row 2 (R2 = R2 + 3R1)
  • Subtract 6 times Row 1 from Row 3 (R3 = R3 - 6R1)
  • Subtract 3 times Row 1 from Row 4 (R4 = R4 - 3R1)
Au' = | 1  3   7  -1 |
      | 0  2  18   2 |
      | 0 -14 -41   7 |
      | 0 -8 -20   4 |

Divide Row 2 by 2:

Au' = |  1   3   7  -1 |
      |  0   1   9   1 |
      |  0 -14 -41   7 |
      |  0  -8 -20   4 |

Now, eliminate entries below the 1 in the second column:

  • Add 14 times Row 2 to Row 3 (R3 = R3 + 14R2)
  • Add 8 times Row 2 to Row 4 (R4 = R4 + 8R2)
Au' = |  1   3   7   -1 |
      |  0   1   9    1 |
      |  0   0  85  21 |
      |  0   0  52  12 |

Multiply Row 4 by 85 and Row 3 by -52 and add them together to make position (4,3) become zero.

Au' = |  1   3    7   -1        |
      |  0   1    9    1        |
      |  0   0   85   21        |
      |  0   0   0  -24/85  |

The determinant of this upper triangular matrix is (1) * (1) * (85) * (-24/85) = -24. We performed one row swap, so:

det(Au) = (-1) * (-24) = 24

5. Solving for 'u' using Cramer's Rule

Finally, we apply Cramer's Rule: u = det(Au) / det(A) = 24 / (-24) = -1

Conclusion

By meticulously applying Cramer's Rule and leveraging the power of row reduction, we've successfully solved for 'u' in the given system of equations. This journey has highlighted the importance of determinants in linear algebra and demonstrated how row reduction can significantly simplify their calculation. The value of 'u' is -1, a testament to the effectiveness of these techniques in unraveling complex systems.

Solve the following system of equations for the variable 'u' using Cramer's Rule. Simplify the determinants using row reduction.

2u + x + y + z = 3
-8u - 7x - 3y + 5z = -3
u + 4x + y + z = 6
u + 3x + 7y - z = 1

Solving for u Using Cramer's Rule A Step-by-Step Guide with Row Reduction