Let A be a 2*3 coefficient matrix ;A
a)If the system Ax=b is consistent is the solution unique? why or why not? explain
b) how many pivots must A have to gurantee that Ax=b has a solution for every vector b in IR^2?
c) can the columns be linearly independent?
4
5-1=4
a) If the system Ax=b is consistent, it means that there exists at least one solution to the equation Ax=b. However, it doesn't necessarily mean that the solution is unique. The uniqueness of the solution depends on the rank of matrix A.
To determine if the solution is unique, we need to find the rank of matrix A. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In this case, A is a 2x3 matrix, which means it has at most 2 linearly independent rows or columns.
If rank(A) = 2, then the solution is unique. This means that there is no redundant information in the system, and every variable has a unique value.
However, if rank(A) < 2, then the solution is not unique. This means that there is some redundancy in the system, and some variables can take multiple values while still satisfying the equation Ax=b.
b) The number of pivots in matrix A determines the number of linearly independent rows or columns it has. In this case, A is a 2x3 matrix, so it can have at most 2 linearly independent rows or columns.
To guarantee that Ax=b has a solution for every vector b in IR^2, we need A to have 2 pivots. Having 2 pivots ensures that there are no redundant rows or columns in the system, and every variable in the equation Ax=b can be uniquely determined.
If A has less than 2 pivots, it means that there is some redundancy in the system, and there are certain vectors b for which the equation Ax=b does not have a solution.
c) The columns of a matrix A can be linearly independent if and only if the rank of A is equal to the number of columns in A.
In this case, A is a 2x3 matrix, so it has 3 columns. If the rank of A is 3, it means that all 3 columns are linearly independent, and there is no linear relationship between them. This implies that the columns are linearly independent.
However, if the rank of A is less than 3, it means that there is some linear relationship between the columns, and they are not linearly independent. This implies that there is some redundancy in the system, and the columns can be expressed as a linear combination of each other.