Let A be a 3 x 3 matrix with two pivot positions. Does the equation Ax=b have at least one solution for every possible b? Explain. Yes. A has a free variable. So the free variable can equal any value such that there is at least one solution for every possible b. O Yes. Since A has three rows and two pivots, there is a row without a pivot. So there is at least one solution for every possible b. O No. A has one free variable, so there will be no solution to the system for any possible b No. Since A has three rows and two pivots, there is a row without a pivot. To have at least one solution for every possible b, all rows of A must have a pivot.

Determine whether the statement below is true or false. Justify the answer. If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row. ***** Choose the correct answer below. O A. This statement is false. If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R™, then the equation Ax = b has a solution for each b in Rm. OB. This statement is true. If A is an mxn matrix and if the equation Ax = b is inconsistent for some b in R™, then the equation Ax = b has no solution for some b in Rm. OC. This statement is false. Since the equation Ax=b has a solution for each b in R™, the equation Ax=b is consistent for each b in Rm. D. This statement is true. If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R™, then the columns of A span Rm

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Determine whether the statement below is true or false. Justify the answer. A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. Choose the correct answer below. O A. This statement is false. If the equation Ax=b has infinitely many solutions, then the vector b cannot be a linear combination of the columns of A. O B. This statement is true. The equation Ax=b is unrelated to whether the vector b is a linear combination of the columns of a matrix A. O C. This statement is true. The equation Ax=b has the same solution set as the equation X₁ a₁ + x₂a₂+...+xa = b. ▶ O D. This statement is false. If the matrix A is the identity matrix, combination of the columns of A. the equation = b has at least one solution, but b is not a linear