Final answer:
The question revolves around constructing augmented matrices for systems of linear equations under different conditions: one with a unique solution, one that is inconsistent, and one involving parameters.
Step-by-step explanation:
To illustrate the concept of systems of linear equations and their augmented matrices, we can provide specific examples that would satisfy the given conditions:
- Unique solution: A system of 5 equations in 3 variables with a unique solution must have three equations that are linearly independent and can determine the variables. An example of an augmented matrix for such a system could be:
\begin{bmatrix}1 & 0 & 0 & | & 5\\0 & 1 & 0 & | & 3\\0 & 0 & 1 & | & -2\\1 & 1 & 1 & | & 6\\2 & -1 & 3 & | & 4\end{bmatrix}
- Inconsistent system: An inconsistent system with 6 equations in 3 variables whose augmented matrix has rank 4 means that there are more equations than necessary and they do not all intersect at a single point. A possible augmented matrix is:
\begin{bmatrix}1 & 2 & 3 & | & 4\\2 & 4 & 6 & | & 8\\0 & 0 & 1 & | & 1\\0 & 0 & 0 & | & 0\\1 & 1 & 1 & | & 2\\1 & 2 & 3 & | & 5\end{bmatrix}
- A system with parameters typically occurs when there are more variables than independent equations. Here's an example of an augmented matrix for a system of 4 equations in 6 variables:
\begin{bmatrix}1 & 0 & 0 & 1 & 2 & 3 & | & 4\\0 & 1 & 0 & 2 & 3 & 4 & | & 5\\0 & 0 & 1 & 3 & 4 & 5 & | & 6\\0 & 0 & 0 & 0 & 0 & 0 & | & 0\end{bmatrix}
The last row, which consists entirely of zeros, indicates that there are free variables, leading to a solution involving parameters.