Simplify the Matrix

Problem

[[1,0,1,0,1,0,1],[0,1,1,0,2,1,2],[0,0,0,1,3,0,3]]

Solution

1. Identify the current form of the matrix. The matrix is already in row echelon form because the leading entries (pivots) of each row are to the right of the leading entries of the rows above them, and all entries below the pivots are zero.

2. Check for reduced row echelon form (RREF) requirements. A matrix is in RREF if it is in row echelon form, every leading entry is 1, and every column containing a leading 1 has zeros everywhere else in that column.

3. Verify the columns containing leading ones. The leading ones are in column 1, column 2, and column 4.

• Column 1 has a 1 at (a_1,1) and zeros in (a_2,1) and (a_3,1)

• Column 2 has a 1 at (a_2,2) and zeros in (a_1,2) and (a_3,2)

• Column 4 has a 1 at (a_3,4) and zeros in (a_1,4) and (a_2,4)

4. Conclude that no further row operations are necessary. Since all conditions for reduced row echelon form are met, the matrix is already fully simplified.

Final Answer

[[1,0,1,0,1,0,1],[0,1,1,0,2,1,2],[0,0,0,1,3,0,3]]=[[1,0,1,0,1,0,1],[0,1,1,0,2,1,2],[0,0,0,1,3,0,3]]

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