The Augmented Matrix
Let $AX=b$ be a system of linear equations where $A$ is an $m\times n$ matrix and $b$ is an $m\times 1$ column vector.
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💡 The augmented matrix of this system is defined as the matrix of size $m \times (n+1)$ whose first $n$ columns are the columns of $A$ and the last column is $b$.
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We denote the augmented matrix by $[A|b]$ and put a vertical line between the first $n$ columns and the last column $b$ while writing it.
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The Gaussian Elimination Method
- Consider the system of linear equations $AX=b$.
- Form the augmented matrix of the system $[A|b]$.
- Perform the same operations on $[A|b]$ that were used to bring $A$ into the reduced row echelon form.
- Let $R$ be the submatrix of the obtained matrix of the first $n$ columns and $c$ be the submatrix of the obtained matrix consisting of the last column.
- We write the obtained matrix as $[R|c]$. Notice that $R$ is the reduced row echelon matrix obtained by row reducing $A$
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- The solution of $AX=b$ are precisely the solutions of $RX=c$.
- Form the corresponding system of linear equations $RX=c$.
- Find ALL the solutions of $RX=c$ and hence of $AX=b$.
Homogeneous System of Linear Equations
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💡 0 is always a solution of a homogeneous system of linear equations $AX=0$. This solution is called Trivial Solution.
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For a homogeneous system, there are only two different possibilities:
- 0 is the unique solution.
- There are infinitely many solutions other than 0.
In a homogeneous system of equations, if there are more variables than equations, then it is guaranteed to have non-trivial solutions.