Gaussian Elimination: Solving Systems of Linear Equations

1. Setting the Stage: From Equations to Matrices

A system of linear equations can be elegantly represented in the matrix form $AX=B$, where:

• $A$ is the coefficient matrix, containing the coefficients of the variables.
• $X$ is a column vector of the variables (e.g., $x,y,z$).
• $B$ is a column vector of the constants on the right-hand side.

To analyze the system, we combine $A$ and $B$ into a single matrix called the augmented matrix, written as $[A|B]$. This matrix contains all the information about the system of equations. Our entire analysis will revolve around this augmented matrix.

2. Consistency: The Rouché–Capelli Theorem

Before we try to solve a system, we must first ask: does a solution even exist? A system is called consistent if it has at least one solution (either one unique solution or infinitely many). If no solution exists, it is inconsistent.

The Rouché–Capelli theorem provides a precise test for consistency by comparing the rank of the coefficient matrix $A$ with the rank of the augmented matrix $[A|B]$.

Here's the complete picture, which you should commit to memory. Let's say the number of variables in the system is $n$.

To summarize the conditions:

1. No Solution (Inconsistent): If $\text{rank}(A)\ne \text{rank}([A|B])$. This happens when the row reduction process creates a row like $[00...|c]$ where $c$ is a non-zero constant. This implies an absurdity like $0=c$, indicating a contradiction.
2. Unique Solution (Consistent): If $\text{rank}(A)=\text{rank}([A|B])=n$, where $n$ is the number of variables. The ranks are equal and "full."
3. Infinite Solutions (Consistent): If $\text{rank}(A)=\text{rank}([A|B])<n$. The ranks are equal but less than the number of variables. This implies there are "free" variables that can be chosen arbitrarily, leading to an infinite set of solutions.

3. The Formal Method: Gaussian Elimination

The most reliable way to determine the ranks and solve the system is Gaussian elimination. This is the exact same process of using elementary row operations to reduce a matrix to row echelon form that you learned in the previous lesson. The only difference is that we apply it to the augmented matrix $[A|B]$.

Let's see this in action with a typical GATE problem involving unknown parameters.

4. Exam Shortcuts: The Logical Approach

While Gaussian elimination is the definitive method, for the time-sensitive GATE exam, you should also develop an intuition for shortcuts. The same video introduces these "logical" methods brilliantly.

No Solution: Look for Contradictions

Often, you can spot an inconsistency by simply manipulating the equations themselves. If you can combine some equations to produce a result that directly contradicts another equation, the system has no solution.

Infinite Solutions: Look for Dependency

If one equation can be formed by a linear combination of others, it provides no new information. This is called a dependent system. Such systems, if consistent, will have infinite solutions.

Unique Solution: Check the Determinant

This is the most common and powerful shortcut. For a system of $n$ equations in $n$ variables (i.e., the coefficient matrix $A$ is square), the condition for a unique solution is $\text{rank}(A)=n$. From our previous lesson, we know this is equivalent to $\det (A)\ne 0$.

If a question asks for the condition for a unique solution for a square system, simply calculate the determinant of the coefficient matrix and set it to be non-zero.

Conclusion

You now have a complete toolkit for analyzing systems of linear equations for the GATE exam.

Key Takeaways:

• Consistency: A system is consistent (solvable) if $\text{rank}(A)=\text{rank}([A|B])$. Otherwise, it is inconsistent (no solution).
• Solution Types:
  • Unique Solution: $\text{rank}(A)=\text{rank}([A|B])=n$ (number of variables).
  • Infinite Solutions: $\text{rank}(A)=\text{rank}([A|B])<n$.
• Gaussian Elimination: The formal method to find ranks and solve the system by reducing the augmented matrix to row echelon form.
• GATE Shortcuts:
  • Unique Solution: For square systems, check if $\det (A)\ne 0$.
  • No Solution: Look for direct contradictions between equations.
  • Infinite Solutions: Look for dependent equations (one being a combination of others).

In our next lesson, we will venture into another core topic of linear algebra: eigenvalues and eigenvectors. The concepts of determinants and rank that you've mastered will be absolutely essential for understanding and calculating these special values and vectors that describe the fundamental properties of linear transformations.