Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 27
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Question A 27 

Which of the following statements are true for all invertible invertible matrices :

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

By isolating one property of an inverse can you use it to deduce other properties that support or eliminate options? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that the definition of an inverse of denoted is where is the identity. Now assume we want to solve for some . If is invertible then by the above property we can left multiply by to get so we conclude that if is invertible then only has , the trivial solution and that option c is correct. Now assume we wanted to find eigenvalues . If we wanted to find zero eigenvalues then we'd need to solve but as we saw above, the only solution to this if is invertible is which is not a valid eigenvector. Therefore, if is invertible then it does not have any zero eigenvalues and option d is correct. A square matrix of size always has eigenvalues. If the matrix has rank then there are always nonzero eigenvalues and zero eigenvalues. Since in this case we have shown that there are no zero eigenvalues then or , thus the matrix has rank and option a is correct. Recall that rank has to do with the number of pivots in a row reduced matrix. If has full rank (rank ) then there will be pivots which means two things. We have shown that all invertible matrices have rank . Firstly this means that the row reduced form of is the identity matrix (since it has pivots) so option b is correct and secondly it means that there are no rows of zeros which means that , so option e is correct. Therefore, from starting with the definition of an inverse we were able to deduce that all of the options are correct. 