Question

Suppose that A is an invertible n by n matrix, with real valued entries.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.

A is row equivalent to the identity matrix I_n.A has fewer than n pivot positions.The equation Ax=0 has only the trivial solution.For some vector b in Rⁿ, the equation Ax=b has no solution.There is an n by n matrix C such that CA=I_n.None of the above.

Answer 1

1. A is row equivalent to the identity matrix I_n. True. If A is invertible, then the columns of A are linearly independent so that A can be row reduced to the identity matrix I_n.
2. A has fewer than n pivot positions. False. A has exactly n pivot positions.
3. The equation Ax=0 has only the trivial solution. True. IF A is invertible, and AX = 0, then X = A^-10 = 0.
4. For some vector b in Rⁿ, the equation Ax=b has no solution. False. If A is invertible, then the columns of A span Rⁿ so that the equation Ax = b is always consistent.
5. There is an n by n matrix C such that CA=I_n. True.