Question

Let A be an mxn matrix. Consider the statement. "For each b in Rm, the equation Ax b has a solution." Because of a fundamental theorem about such matrix equations, this statement is equivalent to what other statements? Choose all that apply. A. The rows of A span Rn. B. Each b in Rm is a linear combination of the columns of A C. The columns of A span R D. The matrix A has a pivot position in each column. E. The matrix A has a pivot position in each row.

Answer 1

Answer: The correct answer is : B. Each b in Rm is a linear combination of the columns of A

C. The columns of A span R

E. The matrix A has a pivot position in each row.

Step-by-step explanation: These conditions correspond to Theorem 4: A square matrix A is invertible, yes and only if the det A ≠ 0

1. If the matrix is not full range, there are infinite solutions

2. Each row must have at least one pivot so that it has a solution

3. When a row is not linearly independent, the matrix is not full range and its determinant is zero.

4. For each b in Rm the equation Ax = b has a solution