how do you write the matrix:
1 2 -1 | 3
7 -1 0 | 2
3 2 1 | -1
in row echelon form?
how do you write the matrix:
1 2 -1 3
7 -1 0 2
3 2 1 -1
in row echelon form?
To write the given matrix in row echelon form, we need to use a sequence of row operations to transform the matrix. The goal is to create a triangular matrix with leading 1's from left to right, and zeros below each leading 1. Here's how you can do it step by step:
Step 1: Swap rows if necessary
Check the first column (also known as the pivot column) and see if there is a row with a non-zero entry in that column. In our case, the first row has a non-zero entry (1) in the first column, so we don't need to swap rows.
Step 2: Scale the pivot row
To make the pivot entry (the first entry in the pivot row) equal to 1, divide the entire pivot row by the value of the pivot entry. In this case, divide the first row by 1.
1 2 -1 | 3
7 -1 0 | 2
3 2 1 | -1
Step 3: Eliminate the entries below the pivot
To eliminate the entries below the pivot, perform row operations to zero out the entries below the pivot. Start with the second row.
Multiply the first row by 7 and subtract it from the second row:
(Row 2) - 7(Row 1) -> (Row 2)
1 2 -1 | 3
0 -15 7 | -19
3 2 1 | -1
Multiply the first row by 3 and subtract it from the third row:
(Row 3) - 3(Row 1) -> (Row 3)
1 2 -1 | 3
0 -15 7 | -19
0 -4 4 | -10
Step 4: Carry out the same steps for the remaining rows
To zero out the entries below the pivot in the second column, perform row operations involving the second row and the third row.
Multiply the second row by -4 and add it to the third row:
(Row 3) - (-4)(Row 2) -> (Row 3)
1 2 -1 | 3
0 -15 7 | -19
0 0 -12 | 26
Step 5: The matrix is now in row echelon form
The matrix is now in row echelon form. All the entries below the leading ones (the pivots) are zeros.
So, the matrix in row echelon form is:
1 2 -1 | 3
0 -15 7 | -19
0 0 -12 | 26