Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 23
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Question A 23 

Let and be matrices and let be a vector. If is an matrix, is a row vector or a column vector, and how many entries does it have? Justify your answer. 
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! 
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Hint 

For matrix multiplication the number of columns of the first matrix must be the same as the number of rows of the second one. 
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.

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can only multiply 2 matrices if the number of columns of is the same as the number of rows of . Thus if the product is a matrix, then we know has size and has size , for some positive integer . If the transposed matrix is size , then the original matrix must be size . This implies that is of size and is of size for some positive integers Now, if were a column vector, then would also be a column vector, of size for some . However, since , we know that must be a row vector. Finally, a row vector is of size for some , and we know from our earlier reasoning is also of size . Thus, for our matrix operations to agree, we must have that

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