Suppose A is nxn and the equation Ax=b has a solution for each bin Rⁿ. Explain why A must be invertible. (Hint: Is A row equivalent to Iₙ?]
Choose the correct answer below.
A. If the equation Ax=b has a solution for each b in Rⁿ, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A follows that A is 1. Therefore, A is invertible.
B. If the equation Ax=b has a solution for each b in Rⁿ, then A does not have a pivot position in each row. Since A is square, and I, is square, A is row equivalent to I. Therefore, A is invertible.
C. If the equation Ax=b has a solution for each b in Rⁿ, then A has one pivot position. It follows that Ais row equivalent to I. Therefore, A is invertible.
D. If the equation Ax = b has a solution for each b in Rⁿ, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A. follows that A is row equivalent to In Therefore, A is invertible.