1. **State the problem:** Determine if the given matrix $$A=\begin{bmatrix}1 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$$ is in Row Echelon Form (REF), Reduced Row Echelon Form (RREF), or neither. 2. **Recall definitions:** - A matrix is in **Row Echelon Form (REF)** if: - All nonzero rows are above any rows of all zeros. - The leading entry of each nonzero row is strictly to the right of the leading entry of the row above it. - Entries below each leading entry are zero. - A matrix is in **Reduced Row Echelon Form (RREF)** if it is in REF and additionally: - The leading entry in each nonzero row is 1. - Each leading 1 is the only nonzero entry in its column. 3. **Check the matrix:** - Row 1 leading entry is 1 at column 1. - Row 2 leading entry is 1 at column 2, which is to the right of column 1. - Row 3 leading entry is 1 at column 4, which is to the right of column 2. - Row 4 is all zeros, below nonzero rows. 4. **Check zeros below leading entries:** - Below leading 1 in row 1, column 1: rows 2,3,4 have 0 in column 1. - Below leading 1 in row 2, column 2: rows 3,4 have 0 in column 2. - Below leading 1 in row 3, column 4: row 4 has 0 in column 4. 5. **Check if leading entries are 1:** All leading entries are 1. 6. **Check if each leading 1 is the only nonzero entry in its column:** - Column 1: leading 1 in row 1, no other nonzero entries. - Column 2: leading 1 in row 2, no other nonzero entries. - Column 4: leading 1 in row 3, no other nonzero entries. - Column 3 has nonzero entries (4 in row 1, 3 in row 2) but no leading 1 in that column. Since columns with leading 1s have zeros elsewhere, but column 3 has nonzero entries and is not a pivot column, this is allowed in REF but not in RREF. **Conclusion:** The matrix is in Row Echelon Form (REF) but not Reduced Row Echelon Form (RREF). **Final answer:** b. REF