Final answer:
The given matrix is not in reduced row-echelon form because the leading 1 in the third row has non-zero entries to its right and the row of all zeros is not at the bottom.
Step-by-step explanation:
The question you've asked is about determining whether a given matrix is in reduced row-echelon form. To checklist the properties of a matrix that is in reduced row-echelon form we look for: 1) If a row has nonzero entries, the first nonzero entry is 1 (called a leading 1); 2) For each leading 1, all other entries in its column are 0; 3) The leading 1 in each row is to the right of the leading 1 in the row above it; 4) Any rows containing only zeros are at the bottom of the matrix. The given matrix has the following form:
[1 0 0 0]
[0 0 0 0]
[0 1 5 1]
If we refer to the properties above, we see that the third row does not satisfy the requirements for reduced row-echelon form because the leading 1 in that row should have all zeros to its right within the same row, but here we have two non-zero entries (5 and 1) to the right of the leading 1. Additionally, we typically expect the row of all zeros (second row) to be at the bottom of the matrix. Therefore, the matrix provided is not in reduced row-echelon form.