Emth-recording on 05 March 2025 at 08.52.46 AM

Matrix Overview

• Number of Variables: 4 (x1, x2, x3, x4)

• Number of Pivots: 3

  • Pivots are the first nonzero value in each row of echelon form.

• Free Variables: 1 (x2 - no pivot in second column)

Gaussian Elimination Process

Initial Setup

• Augmented matrix represents coefficients and constants from equations.

• Matrix structure: Left side - coefficients, Right side - constants.

Performing Row Operations

• Ensure the first pivot is in the top left.

• If zero, perform row swap.

• Use the pivot to make all entries below it zero:

  • Replace row two: Row2 <- Row2 + Row1 (to make below the pivot zero)

  • Replace row three: Row3 <- Row3 - 20 * Row1 (if needed, adjust based on entries)

Pivoting Down

• Move to the next pivot by stepping down right.

• If zero, move the zero row down.

• New pivot is the first nonzero in a new row, repeat the elimination process:

  • E.g. Replace Row3 to make it zero based on the current pivot.

Check for Row Echelon Form

• Each leading entry (pivot) in each row goes further right as you descend the matrix downwards.

• Count pivots and variables:

  • Equal number of pivots and variables indicates a unique solution.

Back Substitution

• Start at the bottom row and work upwards to find variable values:

  • For row 3: Express variable i3, substitute in row 2 for i2, then in row 1 for i1.

  • Maintain notation clarity, avoid confusion with 'equals' marks; use implication signs to show connection.

Special Cases

• If there are more equations than variables (overdetermined system), it might be needed to identify redundant equations - these can be expressed in terms of others.

• Inconsistent system arises if all variables have zero coefficients in the equation row but a non-zero right-hand side constant.

Example Problem Context

• Circuits represented as a network of one-way streets, using equations to model flow at junctions.

• Conservation of mass principle guides the flow rate equations.

• After forming equations, convert to an augmented matrix for Gaussian elimination.

Forming the Augmented Matrix

• Each equation rewrites in standard form, guaranteeing variables on the left and constants on the right.

• Include all necessary zero coefficients in the matrix for proper representation.

Solving Network Flow Problems

• Pivot counting leads to identification of free variables and the potential for infinitely many solutions.

• Conditions must be set on flow values to ensure feasibility in networks, indicating minimum flow requirements.

Summary

• Gaussian elimination is an essential method for solving linear systems.

• Understanding the implications of row operations is crucial in solving for variables and determining the nature of the solutions (unique, infinite, or no solutions).