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Problem 10

# Indicate whether the matrix is in rowreduced form. $$\left[\begin{array}{ll|l}1 & 1 & 3 \\ 0 & 0 & 0\end{array}\right]$$

### Short Answer

Expert verified
The given matrix $$\left[\begin{array}{ll|l}1 & 1 & 3 \\ 0 & 0 & 0\end{array}\right]$$ satisfies all three conditions for row-reduced form: zero rows are at the bottom, pivots are to the right of those in rows above if applicable, and other elements in pivot columns are zero. Therefore, it is in row-reduced form.
See the step by step solution

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## Step 1: Identify the pivot elements

In this matrix, the pivot elements are the first non-zero elements in their respective rows. The first row has a pivot of 1 in column 1, and the second row has no pivot since it is an all-zero row. Matrix with pivots identified: $\left[\begin{array}{ll|l} \textbf{1} & 1 & 3 \\ 0 & 0 & 0 \end{array}\right]$

## Step 2: Check if zero rows are at the bottom

The given matrix has only one row with all zero elements, which is already at the bottom. So, the first condition is satisfied.

## Step 3: Check if pivots are to the right of those in rows above

In this case, there is only one pivot in row 1 and no pivot in row 2. Since there is no pivot in row 2, there is no need to consider this condition. Therefore, this condition is irrelevant for this matrix.

## Step 4: Check if other elements in pivot columns are zero

The pivot element in the first row is 1. According to the third condition, all other elements in the first column must be 0. In this case, since there is only one other element (which is 0) in the first column, this condition is satisfied. #Conclusion#Since the given matrix satisfies all three conditions, it is in row-reduced form.

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