MATH 125 Study Guide - Midterm Guide: Row Echelon Form, Gaussian Elimination, Augmented Matrix
Course CodeMATH 125
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MATH 125 (R1) Winter 2017
Sample Midterm Exam – Solutions
Note: Row reduction of matrices is omitted from the solutions below. In your exam, however, you
should show all you work, indicating which row operations were applied at each step in the process.
1. Solve the following system of linear equations by ﬁrst forming its augmented matrix and then
bringing it to its reduced row echelon form. Give your answer in vector form.
2x2+ 4x3+ 2x4=−2
−x1+ 2x3−4x4= 3
Solution: Applying the row reduction algorithm, we ﬁnd that the reduced row echelon form
of the system is
1 0 −2 4 −3
0 1 2 1 −1
0 0 0 0 0
The leading variables are therefore x1and x2, and the free variables are x3and x4. Assigning
the parametric values sand tto the free variables x3and x4, respectively, and re-writing the
above matrix as a system of equations, we see that the general solution of the system is
x1=−3 + 2s−4t
(s, t ∈R),
or, in vector form,
(s, t ∈R).
2. a) Let A= (2,3) and B= (−1,4) be points in R2. Compute the length ||−→
AB|| of −→
b) Let ~u,~v be orthogonal unit vectors in Rn. Compute the length ||~u +~v|| of ~u +~v.
Solution: a) −→
AB = [−1−2,4−3] = [−3,1], so
AB|| =p(−3)2+ 12=√10.
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