To determine the best description for the given matrix, let's carefully analyze its structure:
[tex]$
\left[\begin{array}{ccc:c}
1 & -4 & 3 & 5 \\
1 & 1 & 9 & -8 \\
0 & 0 & 1 & 9
\end{array}\right]
$[/tex]
### Step-by-Step Analysis:
1. Identify if it is an augmented matrix:
- An augmented matrix includes the coefficients of the variables and the constants from the equations on the right side of the augmented line (usually represented by a colon).
- In the given matrix, there is a partition (represented by the colon) separating the first three columns from the last column. This suggests that the last column represents the constants from the linear equations.
2. Check if the matrix is in row-echelon form:
- Row-echelon form criteria:
- All non-zero rows are above any rows of all zeros.
- The leading entry (first non-zero number from the left) of each non-zero row after the first occurs to the right of the leading entry of the previous row.
- The leading entry in any non-zero row is 1, and it is the only non-zero entry in its column.
- Analyzing each row:
- The first row [tex]\( [1, -4, 3, 5] \)[/tex] has a leading entry of 1.
- The second row [tex]\( [1, 1, 9, -8] \)[/tex] has a leading entry of 1 as well. However, it is not to the right of the leading entry of the first row, violating the row-echelon form criteria.
- The third row [tex]\( [0, 0, 1, 9] \)[/tex] has a leading entry of 1, which is to the right of the leading entry of any non-zero elements in the second row.
Although the third row adheres to the criteria, the second row does not satisfy the requirement that the leading entry must be to the right of the leading entry of the previous row.
From this analysis, we can conclude that:
- The matrix is an augmented matrix because of the presence of the constants after the colon line.
- The matrix is not in row-echelon form due to the second row violating the criteria of row-echelon form.
Therefore, the best phrase that describes the given matrix is:
augmented matrix
So the answer is:
augmented matrix
