Question

Suppose that A is an invertible n by n matrix, with real valued entries. Which of...

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 In.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=In.None of the above.

Homework Answers

Answer #1
  1. A is row equivalent to the identity matrix In. True. If A is invertible, then the columns of A are linearly independent so that A can be row reduced to the identity matrix In.
  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=In. True.
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