2.5 Echelon Form and Solving Linear Systems

This section covers the Echelon Form of a matrix, the rank of a matrix, and methods for solving systems of linear equations including Gaussian elimination and the matrix inversion method.

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Echelon Form of a Matrix

A matrix is in Echelon Form (Row Echelon Form) if:

1. All non-zero rows are above any rows of all zeros.
2. The leading entry (first non-zero entry) of each non-zero row is $1$ (called a pivot).
3. Each pivot is to the right of the pivot in the row above it.
4. All entries below each pivot are $0$.

Reduced Echelon Form (RREF)

A matrix is in Reduced Echelon Form if it satisfies all Echelon Form conditions plus:

• All entries above and below each pivot are $0$.

Example:

$\text{Echelon Form:}\left(\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 0& 0& 1\end{array}\right)\text{RREF:}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

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Rank of a Matrix

The rank of a matrix $A$, denoted $\rho (A)$ or $\text{rank}(A)$, is the number of non-zero rows in its Echelon Form.

Key facts:

• For an $n\times n$ matrix: $0\le \text{rank}(A)\le n$
• If $\text{rank}(A)=n$, the matrix is non-singular (invertible) and $\det (A)\ne 0$.
• If $\text{rank}(A)<n$, the matrix is singular and has no inverse.

Example: If a $3\times 3$ matrix reduces to: $\left(\begin{array}{ccc}1& 2& -1\\ 0& 1& 3\\ 0& 0& 0\end{array}\right)$ then $\text{rank}(A)=2$ (two non-zero rows).

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Finding the Inverse Using Row Operations (Gauss-Jordan Method)

To find $A^{-1}$ for an $n\times n$ matrix $A$:

Step 1: Form the augmented matrix $[A\mid I]$.

Step 2: Apply elementary row operations to reduce the left block to $I$.

Step 3: The right block becomes $A^{-1}$: $[A\mid I]\stackrel{\text{row ops}}{\to }[I\mid A^{-1}]$

Failure condition: If a row of all zeros appears on the left side during reduction, then $\text{rank}(A)<n$, the matrix is singular, and $A^{-1}$ does not exist.

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Solving Non-Homogeneous Systems: $AX=B$

A non-homogeneous system has the form $AX=B$ where $B\ne 0$.

Method 1: Matrix Inversion Method

If $\det (A)\ne 0$, the unique solution is: $X=A^{-1}B$

Example: Solve the system: $x+y+z=6,2x-y+z=3,x+2y-z=2$

Write as $AX=B$: $A=\left(\begin{array}{ccc}1& 1& 1\\ 2& -1& 1\\ 1& 2& -1\end{array}\right),X=\left(\begin{array}{c}x\\ y\\ z\end{array}\right),B=\left(\begin{array}{c}6\\ 3\\ 2\end{array}\right)$

Find $A^{-1}$ using the Gauss-Jordan method, then compute $X=A^{-1}B$.

Method 2: Cramer's Rule

For a $3\times 3$ system $AX=B$ with $\det (A)\ne 0$: $x=\frac{\det (A_1)}{\det (A)},y=\frac{\det (A_2)}{\det (A)},z=\frac{\det (A_3)}{\det (A)}$

where $A_1$, $A_2$, $A_3$ are obtained by replacing the 1st, 2nd, 3rd column of $A$ with $B$ respectively.

Gaussian Elimination (Augmented Matrix Method)

Step 1: Write the augmented matrix $[A\mid B]$.

Step 2: Apply row operations to reduce to Echelon Form.

Step 3: Use back-substitution to find $x$, $y$, $z$.

Consistency check:

• If a row $[000\mid c]$ with $c\ne 0$ appears → system is inconsistent (no solution).
• If a row $[000\mid 0]$ appears → system has infinitely many solutions.

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Solving Homogeneous Systems: $AX=0$

A homogeneous system has the form $AX=0$. It always has the trivial solution $X=0$.

Non-Trivial Solutions

The system $AX=0$ has non-trivial solutions if and only if: $\det (A)=0\iff \text{rank}(A)<n$

Gaussian Elimination for Homogeneous Systems

Step 1: Write the augmented matrix $[A\mid 0]$.

Step 2: Reduce $A$ to Echelon Form using row operations.

Step 3:

• If $\text{rank}(A)=n$: only the trivial solution $X=0$ exists.
• If $\text{rank}(A)<n$: infinitely many non-trivial solutions exist.

Example: Solve $AX=0$ where: $A=\left(\begin{array}{ccc}1& 1& -1\\ 2& -1& 1\\ 3& 0& 0\end{array}\right)$

If $\det (A)=0$, reduce $[A\mid 0]$ to Echelon Form and express the free variables in terms of a parameter $t$.

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Summary Table