The nullspace of a 3 by 4 matrix A is the line through (2,3,1,0)?
a. what is the rank of A and the complete solution to Ax=0?
b. what is the exact row-reduced echelon form R of A?
To determine the rank of matrix A and the complete solution to Ax=0, and to find the exact row-reduced echelon form R of A, we need to perform row operations on the matrix A until it is in reduced row-echelon form.
Let's start with finding the rank:
1. To find the rank of the matrix A, we perform Gaussian elimination on A to obtain its row-echelon form.
a. Begin by row-reducing the matrix A. This involves applying the following row operations:
- Swap rows (if necessary) to bring a non-zero entry to the top of the first column.
- Scale the first row (if necessary) so that the leading entry becomes 1.
- Use row operations to clear all entries below the leading entry in the first column.
- Repeat the above steps for each subsequent column, working column by column from left to right.
b. After performing these row operations, we obtain the row-echelon form of A.
2. The rank of matrix A is equal to the number of non-zero rows in the row-echelon form obtained in step 1. This will give us the rank of A.
Next, let's find the complete solution to Ax=0:
3. Consider the row-echelon form obtained in step 1.
a. Identify the positions of the leading entries (1s) in each row. These positions correspond to the pivot variables in the system of equations.
b. Express all non-pivot variables (free variables) in terms of the pivot variables.
c. Write the complete solution to Ax=0, using these expressions.
Finally, let's find the exact row-reduced echelon form R of A:
4. To obtain the row-reduced echelon form R of matrix A, continue performing row operations on the row-echelon form obtained in step 1 until each leading entry is the only non-zero entry in its column.
By following these steps, you can determine the rank of A, the complete solution to Ax=0, and the exact row-reduced echelon form R of A.
To find the rank of matrix A and the complete solution to Ax=0, we need to perform row operations on matrix A until it is in its row-reduced echelon form (R).
Given that the nullspace of A is the line through (2,3,1,0), we can conclude that the rank of A is 3 because the nullspace is one-dimensional.
To determine the row-reduced echelon form (R) of A, we first need to construct the augmented matrix [A|0]. Since A is a 3x4 matrix, the augmented matrix will have 3 rows and 5 columns.
Let's perform the row operations step-by-step:
1. Starting with the augmented matrix [A|0], swap rows to ensure that the first non-zero entry in the first column is 1.
[2, 3, 1, 0, 0
... ]
2. Divide the first row by 2 to make the leading entry 1.
[1, 3/2, 1/2, 0, 0
... ]
3. Use row operations to eliminate the entries below the leading 1 in the first column.
[1, 3/2, 1/2, 0, 0
0, ..., ..., ..., ...
0, ..., ..., ..., ...]
4. Use row operations to eliminate the entries above and below the leading 1 in the second column.
[1, 0, -1/2, 0, 0
0, 1, 1/3, 0, 0
0, ..., ..., ..., ...]
5. Use row operations to eliminate the entries above and below the leading 1 in the third column.
[1, 0, -1/2, 0, 0
0, 1, 1/3, 0, 0
0, 0, ..., ..., ...]
Now, the matrix is in its row-reduced echelon form R. From this, we can conclude that:
a) The rank of A is 3 and the complete solution to Ax=0 can be represented as:
x = t * [-1/2, -1/3, 1, 0], where t is a scalar.
b) The exact row-reduced echelon form R of A is:
[1, 0, -1/2, 0, 0
0, 1, 1/3, 0, 0
0, 0, ..., ..., ...]