Question

Find matrices A, not necessarily square, for which the number of solutions to Ax = b has the following properties. Explain your answers.

One solution, regardless of b.
a) Matrix A is invertible
b) Matrix A is singular
c) Matrix A has infinitely many solutions
d) Matrix A has no solution

Answer 1

To find matrices A for which the number of solutions to Ax = b has specific properties, we need to analyze the properties of matrix A. If A is invertible, there will always be one solution. If A is singular, the number of solutions depends on b. If A has infinitely many solutions, the columns of A are linearly dependent. If A has no solution, the columns of A are linearly independent.

Step-by-step explanation:

To find matrices A for which the number of solutions to Ax = b has specific properties, we need to analyze the properties of matrix A.

1. If matrix A is invertible, it means it has a unique inverse. In this case, regardless of the vector b, there will always be one solution to the equation Ax = b.
2. If matrix A is singular, it means it does not have an inverse. In this case, the number of solutions will depend on b. If b is in the column space of A, there will be infinitely many solutions. If b is outside the column space of A, there will be no solution.
3. If matrix A has infinitely many solutions, it means the columns of A are linearly dependent, creating a redundancy in the system. In this case, any vector b in the column space of A will have infinitely many solutions.
4. If matrix A has no solution, it means the columns of A are linearly independent and the system is inconsistent. Any vector b outside the column space of A will have no solution.