Questions: Problems For Problems 1-8, determine whether the given matrices are in reduced row-echelon form, row-echelon form but not reduced row-echelon form, or neither. 1. [0 1; 1 0]. 2. [1 1; 0 0]. 3. [1 0 2 5; 1 0 0 2; 0 1 1 0]. 4. [1 0 -1 0; 0 0 1 2; 0 0 0 0]. 5. [1 0 1 2; 0 0 1 1; 0 0 0 1; 0 0 0 0].

Problems
For Problems 1-8, determine whether the given matrices are in reduced row-echelon form, row-echelon form but not reduced row-echelon form, or neither.
1. [0 1; 1 0].
2. [1 1; 0 0].
3. [1 0 2 5; 1 0 0 2; 0 1 1 0].
4. [1 0 -1 0; 0 0 1 2; 0 0 0 0].
5. [1 0 1 2; 0 0 1 1; 0 0 0 1; 0 0 0 0].

Transcript text: Problems For Problems 1-8, determine whether the given matrices are in reduced row-echelon form, row-echelon form but not reduced row-echelon form, or neither. 1. $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$. 2. $\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$. 3. $\left[\begin{array}{llll}1 & 0 & 2 & 5 \\ 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 0\end{array}\right]$. 4. $\left[\begin{array}{rrrr}1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0\end{array}\right]$. 5. $\left[\begin{array}{llll}1 & 0 & 1 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right]$.

Solution

Solution Steps

To determine whether a matrix is in reduced row-echelon form (RREF), row-echelon form (REF) but not RREF, or neither, we need to check the following conditions:

Row-Echelon Form (REF):
- All nonzero rows are above any rows of all zeros.
- The leading entry of each nonzero row (after the first) occurs to the right of the leading entry of the previous row.
- The leading entry in any nonzero row is 1.
Reduced Row-Echelon Form (RREF):
- The matrix is in REF.
- The leading 1 in each row is the only nonzero entry in its column.

We will check these conditions for each of the given matrices.

Step 1: Analyze Matrix 1

The first matrix is: \[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

Row-Echelon Form (REF) Check:
- The leading entry of the first row is in the second column, and the leading entry of the second row is in the first column. This does not satisfy the condition that the leading entry of each nonzero row (after the first) must occur to the right of the leading entry of the previous row.
Reduced Row-Echelon Form (RREF) Check:
- Since the matrix is not in REF, it cannot be in RREF.

Thus, the matrix is neither in REF nor RREF.

Step 2: Analyze Matrix 2

The second matrix is: \[ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \]

Row-Echelon Form (REF) Check:
- The leading entry of the first row is 1, and the second row is all zeros, which is allowed in REF. Therefore, this matrix satisfies the conditions for REF.
Reduced Row-Echelon Form (RREF) Check:
- The leading 1 in the first row is not the only nonzero entry in its column, so it does not satisfy the conditions for RREF.

Thus, the matrix is in row-echelon form but not reduced row-echelon form.

Step 3: Analyze Matrix 3

The third matrix is: \[ \begin{bmatrix} 1 & 0 & 2 & 5 \\ 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 0 \end{bmatrix} \]

Row-Echelon Form (REF) Check:
- The leading entry of the first row is in the first column, the leading entry of the second row is also in the first column, which violates the condition that the leading entry of each nonzero row (after the first) must occur to the right of the leading entry of the previous row.
Reduced Row-Echelon Form (RREF) Check:
- Since the matrix is not in REF, it cannot be in RREF.

Thus, the matrix is neither in REF nor RREF.