find the row rank and the column rank of the following matrix | 1 2 3 2 1|
| 3 1 -5 -2 1|
| 7 8 -1 2 5|
To find the row rank and column rank of a matrix, we need to perform row and column operations to bring the matrix into its row-echelon form or reduced row-echelon form. The row rank is the number of nonzero rows in the reduced row-echelon form, and the column rank is the number of leading 1's in the reduced row-echelon form.
Let's work through the given matrix step by step:
Starting matrix:
```
| 1 2 3 2 1 |
| 3 1 -5 -2 1 |
| 7 8 -1 2 5 |
```
Step 1: Subtract 3 times the first row from the second row:
```
| 1 2 3 2 1 |
| 0 -5 -14 -8 -2 |
| 7 8 -1 2 5 |
```
Step 2: Subtract 7 times the first row from the third row:
```
| 1 2 3 2 1 |
| 0 -5 -14 -8 -2 |
| 0 -6 -22 -12 -2 |
```
Step 3: Multiply the second row by -1/5 to make the leading coefficient 1:
```
| 1 2 3 2 1 |
| 0 1 2 8/5 2/5 |
| 0 -6 -22 -12 -2 |
```
Step 4: Add 6 times the second row to the third row:
```
| 1 2 3 2 1 |
| 0 1 2 8/5 2/5 |
| 0 0 -10 18/5 8/5 |
```
Step 5: Divide the third row by -10 to make the leading coefficient 1:
```
| 1 2 3 2 1 |
| 0 1 2 8/5 2/5 |
| 0 0 1 -9/10 -4/10 |
```
Step 6: Subtract 2 times the third row from the second row:
```
| 1 2 3 2 1 |
| 0 1 0 26/10 6/10 |
| 0 0 1 -9/10 -4/10 |
```
Step 7: Subtract 3 times the third row from the first row:
```
| 1 2 0 29/10 9/10 |
| 0 1 0 26/10 6/10 |
| 0 0 1 -9/10 -4/10 |
```
The reduced row-echelon form of the matrix is:
```
| 1 2 0 29/10 9/10 |
| 0 1 0 26/10 6/10 |
| 0 0 1 -9/10 -4/10 |
```
The row rank is 3 (all three rows are nonzero), and the column rank is also 3 (there are three leading 1's in the reduced row-echelon form).
