Question

Let A be a 3 times ×2 matrix. Explain why the equation Ax=b cannot be consistent for all b in set of real numbers R3.

Generalize your argument to the case of an arbitrary A with more rows than columns.
Why is the equation Ax=b not consistent for all b in set of real numbers R3​?
When written in reduced row echelon​ form, any 3 ×2 matrix will have at least one column of all zeros. Since there is not a pivot in every​ column, the matrix cannot be consistent.

Answer 1

The equation Ax=b cannot be consistent for all b in the set of real numbers R3 because when a 3×2 matrix is in reduced row echelon form, it will always have at least one column of all zeros. This means that there is not a pivot in every column, resulting in an inconsistent system of equations.

Step-by-step explanation:

When written in reduced row echelon form, any 3×2 matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent for all b in the set of real numbers R3. This means that the equation Ax=b will not have a solution for every b in R3.

This argument can be generalized to any arbitrary matrix A with more rows than columns. In this case, when the matrix is in reduced row echelon form, there will still be at least one column of all zeros. Therefore, the equation Ax=b will not be consistent for all b in R3.