Let A [ a1, a2, a3] be a 3x3 non-singular matrix, where
1 a ,[a1, a2, a3] are the three columns of A. Define a 3 x 4
matrix B by
B[ 2a1+4a2-2a3 , -a1 -4a2 +3a3 ,a2-a3 , 3a1-2a2+6a3]
Show that the system of linear equations Bx= b is consistent for every
3 x 1 matrix b.'
how can i solve it
IDK how to solve it, sry.
To determine whether the system of linear equations Bx = b is consistent for every 3x1 matrix b, we need to check if the augmented matrix [B | b] has a unique solution or infinitely many solutions.
To solve this problem, we'll first compute the augmented matrix [B | b]. Then, we'll perform row operations to bring the augmented matrix into row-echelon form (REF). Finally, we'll analyze the resulting matrix to determine if it has a unique solution or infinitely many solutions.
Let's proceed with the calculation step-by-step:
Step 1: Compute the augmented matrix [B | b]
The augmented matrix is obtained by appending the matrix b as an additional column to matrix B. The resulting augmented matrix is:
[B | b] = [[2a1 + 4a2 - 2a3, -a1 - 4a2 + 3a3, a2 - a3, 3a1 - 2a2 + 6a3] | [b1, b2, b3]]
Step 2: Perform row operations to bring the augmented matrix into row-echelon form (REF)
Apply row operations to transform the augmented matrix [B | b] into row-echelon form (REF). The specific sequence of row operations depend on the entries of the matrix B and the values of b.
Step 3: Analyze the resulting matrix
Once the matrix [B | b] is in row-echelon form (REF), examine the rightmost column (the column containing the values of b) and check if there are any leading 1's (the leftmost non-zero entry in each row) in that column.
If there are no rows with leading 1's in the rightmost column, then the system of linear equations Bx = b is consistent for every 3x1 matrix b. This means that every vector b has a solution.
If there is a row with a leading 1 in the rightmost column, then the system of linear equations Bx = b is inconsistent for some specific 3x1 matrix b. This means that there are certain vectors b for which there is no solution.
By following these steps and analyzing the resulting matrix, you can determine whether the system of linear equations Bx = b is consistent for every 3x1 matrix b.