Inverse Operations: Each elementary operation has an inverse operation.

Definition 2.3.2 (Equivalent Linear Systems): Two linear systems are equivalent if one can be obtained from the other by a finite number of elementary operations.

Lemma 2.3.3: If $Cx=d$ is obtained from $Ax=b$ by an elementary operation, then $Ax=b$ and $Cx=d$ have the same set of solutions.

Theorem 2.3.4: Two equivalent systems have the same set of solutions.

2.3.1 Gauss Elimination Method

Definition 2.3.5 (Elementary Row Operations):
1. Interchange $i^{th}$ and $j^{th}$ rows ($R_{ij}$).
2. Multiply $k^{th}$ row by $c \neq 0$ ($R_k(c)$).
3. Replace $k^{th}$ row by $k^{th}$ row plus $c$ times $j^{th}$ row ($R_{kj}(c)$).
Exercise 2.3.6: Find inverse row operations.
Definition 2.3.7 (Row Equivalent Matrices): Two matrices are row-equivalent if one can be obtained from the other by a finite number of elementary row operations.
Definition 2.3.9 (Forward/Gauss Elimination Method): Method of solving $Ax=b$ by bringing augmented matrix $[A \mid b]$ to upper triangular form.

2.4 Row Reduced Echelon Form of a Matrix

Definition 2.4.1 (Row Reduced Form): A matrix $C$ is in row reduced form if:
1. First non-zero entry in each row is 1.
2. Column containing this 1 has all other entries zero.
Definition 2.4.3 (Leading Term, Leading Column): For a row-reduced matrix, the first non-zero entry of any row is a leading term. Columns containing leading terms are leading columns.
Definition 2.4.4 (Basic, Free Variables): In $Ax=b$, variables corresponding to leading columns in first $n$ columns of row-reduced $[C \mid d]$ are basic variables. Others are free variables.
Remark 2.4.5: If there are $r$ non-zero rows in row-reduced form, there are $r$ leading terms, $r$ basic variables, and $n-r$ free variables.

2.4.1 Gauss-Jordan Elimination

Definition 2.4.6 (Row Reduced Echelon Form): A matrix $C$ is in row reduced echelon form if:
1. It is already in row reduced form.
2. Rows consisting of all zeros come below all non-zero rows.
3. Leading terms appear from left to right in successive rows.
Method to get row-reduced echelon form: Steps 1-5 (detailed in original document).
Remark 2.4.10: Row reduction involves only row operations and proceeds from left to right.
Theorem 2.4.11: The row reduced echelon form of a matrix is unique.

2.4.2 Elementary Matrices

Definition 2.4.13 (Elementary Matrix): A square matrix $E$ of order $n$ obtained by applying exactly one elementary row operation to $I_n$.
Remark 2.4.14: Three types of elementary matrices correspond to the three elementary row operations ($R_{ij}$, $R_k(c)$, $R_{kj}(c)$).
Left multiplication by an elementary matrix performs a row operation.
Definition 2.4.16 (Elementary Column Operations): Column transformations obtained by right multiplication of elementary matrices.

2.5 Rank of a Matrix

Definition 2.5.1 (Consistent, Inconsistent): A linear system is consistent if it admits a solution, inconsistent otherwise.
Definition 2.5.2 (Row rank of a Matrix): The number of non-zero rows in the row reduced form of a matrix. Denoted $\text{row-rank}(A)$.
Remark 2.5.4: $\text{row-rank}(A) \le \text{row-rank}([A \mid b])$.
Remark 2.5.5