In solving simultaneous equations by Gauss - Jordan method, the coefficient matrix is reduced to

To reduce a coefficient matrix using the Gauss-Jordan method, one typically solves a system of linear equations by performing a series of row operations until the matrix reaches reduced row echelon form (RREF). This involves transforming the augmented matrix of the system until we have a matrix that clearly shows the solutions to the equations.

Breakdown of the Problem

1. Identify the Coefficient Matrix: Start with a matrix that consists of the coefficients of the variables in the system of equations.
2. Set Up the Augmented Matrix: Combine the coefficient matrix with the constants from the equations into an augmented matrix.

Relevant Concepts

3. Row Operations: Perform the following three types of elementary row operations:

   • Swap two rows.
   • Multiply a row by a non-zero scalar.
   • Add or subtract the multiple of one row to another row.

4. Aim for RREF: The goal is to transform the matrix such that:

   • Each leading entry of a non-zero row is 1.
   • Each leading 1 is the only non-zero entry in its column.
   • The leading 1s move to the right as you go down the rows.

Analysis and Detail

5. Step-by-step Transformation: Start applying the row operations with the aim of achieving the above conditions. This might involve:
   • Identifying pivot positions.
   • Normalizing rows if required (e.g., making a row's leading coefficient 1 by dividing the entire row by that coefficient).
   • Eliminating other entries in the pivot columns by appropriately adding or subtracting rows.

Verify and Summarize

6. Check the RREF Compliance: After obtaining the final form, verify if it meets the conditions of reduced row echelon form.
7. Summarize Findings: The reduced form will help in easily reading off the solutions to the variables in the system of equations. If the system is consistent and has a unique solution, the last column of the augmented matrix will give you the values directly.

Final Answer

The coefficient matrix, when reduced by the Gauss-Jordan method, will generally showcase a clear structure enabling easy identification of the solution, taking the form:

$\begin{pmatrix} 1 & 0 & 0 & | & x_1 \\ 0 & 1 & 0 & | & x_2 \\ 0 & 0 & 1 & | & x_3 \\ \vdots & \vdots & \vdots & \vdots \end{pmatrix}$

where $x_1, x_2, x_3$ are the solutions of the original equations.