Answer:
Step-by-step explanation:
Breakdown of row-echelon forms of augmented matrices and what they represent about linear systems:
a. Inconsistent System
* Row-Echelon Form: A row exists with all zeros on the left side of the vertical line and a nonzero number on the right side.
* Meaning: This represents an equation like 0 = [nonzero number], which is impossible and indicates the system has no solutions.
b. Consistent System
* Row-Echelon Form: Might look like either of these cases:
* Unique Solution: All leading coefficients (the first nonzero number in each row) are 1. Other entries in the columns with leading coefficients will be 0.
* Infinite Solutions: One or more rows of zeros exist at the bottom of the matrix, indicating free variables.
* Meaning: A consistent system has at least one solution. An infinite number of solutions exist if there are free variables.
