Questions: Question 3 1 pts Which of the following matrices could be expressed in reduced row echelon form by only multiplying only one row by one coefficient? Select all that apply. [2 0 sqrt(8); 0 1 0] [1 0 -2; 0 3/4 8] [0 1 0; -1 0 0] [-1 0 5.2; 0 1 0] [-11 0 3.6; 0 1 31]

Solution Steps

To determine which matrices can be expressed in reduced row echelon form (RREF) by only multiplying one row by a coefficient, we need to check if any of the matrices are already in RREF except for a single row that can be scaled to meet the RREF criteria. A matrix is in RREF if:

1. The leading entry in each nonzero row is 1.
2. Each leading 1 is the only nonzero entry in its column.
3. The leading 1 in a row is to the right of the leading 1 in the row above it.
4. Any rows of all zeros are at the bottom.

We need to determine if any of the given matrices can be expressed in reduced row echelon form (RREF) by multiplying only one row by a coefficient. The matrices provided are:

1. \(\begin{bmatrix} 2 & 0 & \sqrt{8} \\ 0 & 1 & 0 \end{bmatrix}\)
2. \(\begin{bmatrix} 1 & 0 & -2 \\ 0 & \frac{3}{4} & 8 \end{bmatrix}\)
3. \(\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \end{bmatrix}\)
4. \(\begin{bmatrix} -1 & 0 & 5.2 \\ 0 & 1 & 0 \end{bmatrix}\)
5. \(\begin{bmatrix} -11 & 0 & 3.6 \\ 0 & 1 & 31 \end{bmatrix}\)

We evaluate each matrix against the RREF criteria:

1. The leading entry in each nonzero row must be \(1\).
2. Each leading \(1\) must be the only nonzero entry in its column.
3. The leading \(1\) in a row must be to the right of the leading \(1\) in the row above it.
4. Any rows of all zeros must be at the bottom.

After evaluating all matrices, we find that none of them can be converted to RREF by scaling just one row. The results are as follows:

• Matrix 1: Not RREF
• Matrix 2: Not RREF
• Matrix 3: Not RREF
• Matrix 4: Not RREF
• Matrix 5: Not RREF

Final Answer