201 Quiz 1 THUR solution - 110.201 Linear Algebra 1st Quiz Solution Problem 1 Given the following system of equations x 3y z = 1 x y 2z = 14 find all

# Linear Algebra with Applications (3rd Edition)

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110.201 Linear Algebra 1st Quiz, Solution February 15, 2005 Problem 1 Given the following system of equations: x - 3 y + z = 1 x + y + 2 z = 14 find all solutions using Gauss-Jordan elimination procedure. Is this an example of consistent system? Why? Solution 1 - 3 1 1 1 2 1 14 Row(II)-Row(I): 1 - 3 1 0 4 1 1 13 Row(II) ÷ 4: 1 - 3 1 0 1 1 4 1 13 4 Row (I)+ Row (II) × 3: 1 0 7 4 0 1 1 4 43 4 13 4 1
Therefore, the answer is: x = 43 4 - 7 4 z y = 13 4 - 1 4 z z = any real number Of course, consistent system and has infinite solutions. Problem 2 Find the rank of the following matrix 1 0 1 1 2 - 1 1 1 0 0 0 1 1 1 1 1 0 1 1 2 Solution 1 0 1 1 2 - 1 1 1 0 0 0 1 1 1 1 1 0 1 1 2 +(I) -(I) then: 1 0 1 1 2 0 1 2 1 2 0 1 1 1 1 0 0 0 0 0 -(II) then: 1 0 1 1 2 0 1 2 1 2 0 0 - 1 0 - 1 0 0 0 0 0 × ( - 1) then: 1 0 1 1 2 0 1 2 1 2 0 0 1 0 1 0 0 0 0 0 - (III) - (2 × (III)) 2
then: 1 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 Here, we see, the reduced row-echelon form has 3 leading 1’s, therefore, the rank is 3.