Solve The System Of Linear Equations Using Gauss-Jordan

Solve The System Of Linear Equations Using The Gauss Jordan Elimin

Solve the system of linear equations, using the Gauss-Jordan elimination method. A) B) C) D) E)

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Gauss-Jordan elimination is a systematic method used to solve systems of linear equations by transforming the system's augmented matrix into reduced row echelon form (RREF). This process involves a series of elementary row operations—row swapping, scaling rows, and adding multiples of one row to another—to simplify the matrix such that the solutions can be directly read off or easily calculated. The method is particularly valued for its clarity and direct approach in solving small to moderate systems of equations (Lay, 2012).

To illustrate, consider the general system of three equations:

a11x + a12y + a13z = b1

a21x + a22y + a23z = b2

a31x + a32y + a33z = b3

This system can be represented as an augmented matrix:

| a11 a12 a13 | b1 |


| a21 a22 a23 | b2 |


| a31 a32 a33 | b3 |

The Gauss-Jordan method involves creating zeros below and above the leading 1s (pivots) in each column, ultimately arriving at a form where each leading coefficient is 1 and all other entries in the pivot columns are zeros. From this form, solutions can be obtained by back substitution.

For instance, an example calculation may involve the following steps:

  1. Identify the leftmost pivot and make it equal to 1 by dividing its row by the pivot element.
  2. Use row operations to create zeros in all entries below and above the pivot.
  3. Move to the next pivot position and repeat the process.
  4. Once in RREF, interpret the solution directly: the last column variables can be expressed explicitly.

Applying this method ensures that each variable’s value is determined clearly, provided the system is consistent and has a unique solution. If the matrix results in a row of zeros with a non-zero entry in the augmented part, the system has no solution; if there are free variables, infinitely many solutions exist (Holliday, 2013).

In conclusion, Gauss-Jordan elimination offers a straightforward, algorithmic approach to solving linear systems, making it highly effective for both manual calculations and computational algorithms in linear algebra applications.

References

  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Holliday, J. (2013). Introduction to Linear Algebra. McGraw-Hill Education.

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