Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 04
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Question A 04 

Consider the matrix given by For what and does the matrix have rank 2? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What does it mean for the matrix to have rank 2? How might computing its row echelon form help you? 
Hint 2 

For a rank 2 matrix in row echelon form, the first two rows must each have at least 1 nonzero entry, while the last row must consist of zeroes only. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. We reduce the matrix to row echelon form. Subtracting the first row from the last row transforms into Next, subtracting times the second row from the third row yields Clearly, this matrix has rank at least 2 (since, for instance, the first and third columns are linearly independent regardless of the value of ). On the other hand, it must have rank less than 3 whenever or (or both), which is to say that has rank 2 exactly when or (or both). 