reduce echolon form
| 1 3 5 6|
| 4 1 -2 4|
| -2 0 3 1|
To reduce the given matrix to its row-echelon form, we'll perform a series of row operations. The goal is to get zeros below the main diagonal in each column, and to have leading coefficients of each row be 1. Here's the step-by-step reduction:
Starting matrix:
```
| 1 3 5 6 |
| 4 1 -2 4 |
| -2 0 3 1 |
```
Step 1: Subtract 4 times the first row from the second row:
```
| 1 3 5 6 |
| 0 -11 -22 -20 |
| -2 0 3 1 |
```
Step 2: Add 2 times the first row to the third row:
```
| 1 3 5 6 |
| 0 -11 -22 -20 |
| 0 6 13 13 |
```
Step 3: Multiply the second row by -1/11 to make the leading coefficient 1:
```
| 1 3 5 6 |
| 0 1 2 2 |
| 0 6 13 13 |
```
Step 4: Subtract 3 times the second row from the first row:
```
| 1 0 -1 0 |
| 0 1 2 2 |
| 0 6 13 13 |
```
Step 5: Subtract 6 times the second row from the third row:
```
| 1 0 -1 0 |
| 0 1 2 2 |
| 0 0 1 1 |
```
Step 6: Add the third row to the first row:
```
| 1 0 0 1 |
| 0 1 2 2 |
| 0 0 1 1 |
```
Step 7: Subtract the third row from the second row:
```
| 1 0 0 1 |
| 0 1 0 1 |
| 0 0 1 1 |
```
This is now in row-echelon form. The reduced matrix is:
```
| 1 0 0 1 |
| 0 1 0 1 |
| 0 0 1 1 |
```
So, the reduced row-echelon form of the given matrix is:
```
| 1 0 0 1 |
| 0 1 0 1 |
| 0 0 1 1 |
```
