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2004Spring20MidtermI-sol

2004Spring20MidtermI-sol - mt1.nb 1 Midterm Problems...

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Midterm Problems Problem 1 Let In[1]:= A = J 1 2 - 1 - 1 N B = J 3 0 2 1 N Out[1]= 88 1,2 < , 8 - 1, - 1 << Out[2]= 88 3,0 < , 8 2,1 << Find In[6]:= 3 A + 2 B êê MatrixForm A.B êê MatrixForm A.A.Transpose @ B D êê MatrixForm Out[6]//MatrixForm= J 9 6 1 - 1 N Out[7]//MatrixForm= J 7 2 - 5 - 1 N Out[8]//MatrixForm= J - 3 - 2 0 - 1 N Problem 2 Find the parametric form to the general solution to the system of linear equations 2 x 1 + 4 x 2 + x 4 = 2; x 1 + 2 x 2 + x 3 + 2 x 4 = 4; 3 x 1 + 6 x 2 + 2 x 3 + x 4 = 4; Solution. The augmented matrix of this system of linear equations is In[11]:= M2 = i k j j j j j j j 2 4 0 1 2 1 2 3 2 4 3 6 2 1 2 y { z z z z z z z Out[11]= 88 2,4,0,1,2 < , 8 1, 2,3, 2,4 < , 8 3, 6,2, 1,2 << mt1.nb 1
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which has reduced row echelon form In[12]:= RowReduce @ M2 D êê MatrixForm Out[12]//MatrixForm= i k j j j j j j j 1 2 0 0 0 0 0 1 0 0 0 0 0 1 2 y { z z z z z z z This means the general solution takes the form x 4 = 2; x 3 = 0 x 2 isfree x 1 = - 2 x 2 Another way to write this is i k j j j j j j j j j j j j z 1 z 2 z 3 x 4 y { z z z z z z z z z z z z = i k j j j j j j j j j j j j - 2 x 2 z 2 0 2 y { z z z z z z z z z z z z = i k j j j j j j j j j j j j 0 0 0 2 y { z z z z z z z z z z z z + z 2 i k j j j j j j j j j j j j - 2
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