c. If A is an n xn matrix, then the equation Ax b has at least one solution for

Question

c. If A is an n xn matrix, then the equation Ax b has at least one solution for each b in Rn A. True, by the Invertible Matrix Theorem Ax= b has at least one solution for each b in Rn for all matrices of size n × n. B. True; by the Invertible Matrix Theorem if Ax b has at least one solution for each b in Rn, then the matrix is not invertible O C. False; by the Invertible Matrix Theorem Ax - b has at least one solution for each b in IR" only if a matrix is invertible O D. False; by the Invertible Matrix Theorem if Ax b has at least one solution for each b in R", then the equation Ax - b has no solution d. If the equation Ax 0 has a nontrivial solution, then A has fewer than n pivot positions A. False, by the Invertible Matrix Theorem f the equation Ax= has a no trivial solution then the columns o do not om a nea independent set here re has vot positions. O B. True; by the Invertible Matrix Theorem if the equation Ax 0 has a nontrivial solution, then the columns of A form a linearly independent set. Therefore, A has fewer than n pivot positions C. False: by the Invertible Matrix Theorem if the equation Ax-0 has a nontrivial solution, then matrix A is invertible. Therefore, A has n pivot positions. O D. True; by the Invertible Matrix Theorem if the equation Ax 0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions.

Explanation / Answer

c) Option A. It is true by invertible theorem there exist atleast one x such that x = A-1b.

d) Option B is true.

e) Option C is true

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