Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (a)
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Question 02 (a) 

Let
The matrices A and B are related by , where M is an invertible 4x4 matrix. a) Find the rank and a basis of the row space of A. 
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 

How does left multiplication of a matrix by a invertible matrix change the rank and row space? What are properties preserved under those changes? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the matrix A and B are related by the left multiplication of that invertible matrix M, they share many characteristic, including rank and they row space. The reason being that the left multiplication by M can be understood as simply being a set of (invertible) row operations and the rank and row space are properties that are preserved under (invertible) row operations. Now, as B is in row echelon form, it is easier to read this information from the matrix B, so we will use to answer the question. As B has 3 pivots, we see that the rank is 3. We can also take each row that has a pivot to form a basis of the row space, thus our basis is given by:
