21-06-2024 17:19

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Tags : mathematics linear algebra Matrix Linear Equations

Methods to check consistency

Certainly! Let’s delve into the methods for testing the consistency of a system of linear equations. I’ll break down the steps for both homogeneous and non-homogeneous systems.

Method 1: Square Matrix (Applicable for n unknowns and n equations)

1. Square Matrix A: If you have a system of n equations with n unknowns, the matrix of coefficients (A) must be square. In other words, the number of equations should match the number of unknowns.

   Consider the system:

   $$\begin{gathered} AX=B \\ AX=B \end{gathered}$$

   • If A is a square matrix, you can proceed with the following steps.

   • Multiply both sides by the inverse of A:

     $A^{-1}AX=A^{-1}BA-1AX=A-1B$

     This simplifies to:

     $X=A^{-1}BX=A-1B$

Method 2 : Non-Homogeneous Linear Equations:

1. Matrix Form: Write down the given system in matrix form:

   $AX=BAX=B$

2. Row Echelon Form: Reduce matrix A to row echelon form.

3. Rank Comparison:

   • Determine the rank of matrix A.
   • Calculate the rank of the augmented matrix $([A|B]).$

4. Consistency Check:

   • If $Rank(A)<Rank(([A|B]))$, the system is inconsistent (no solution).
   • If $Rank(A)=Rank(([A|B]))$:
     • If the rank equals the number of unknowns, there is a unique solution.
     • Otherwise, there are infinitely many solutions.

Method 3: Homogeneous System:

1. Matrix Form: Write down the given system in matrix form:

   $AX=0AX=0$

2. Row Echelon Form: Reduce matrix A to row echelon form.

3. Rank Comparison:

   • Compare the rank of matrix A with the number of unknowns.

4. Solution Types:

   • If $Rank(A)=$number of unknowns, the only solution is the trivial solution (zero solution).
   • If $Rank(A)<$ number of unknowns, there are non-trivial solutions (parameters take arbitrary values).

References