Science:Math Exam Resources/Courses/MATH152/April 2016/Question A 20

MATH152 April 2016

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Question A 20
If A is a matrix with 5 rows and 4 columns such that the set of solutions to the homogeneous system $A{\textbf {x}}=0$ has 2 parameters, what is the rank 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
Consider the Rank-Nullity Theorem.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The set of solutions to the homogeneous system has $2$ parameters, so the dimension of the null space of this matrix is $2$. The dimension of the domain is $4$. By the Rank-Nullity Theorem, we have $\mathrm {rank} (A)+\mathrm {null} (A)=4$. Thus, the rank of $A$ is $\color {blue}2$.