sol1 - Mathematics 20F Fall 2008 Quiz 1 Name: No aids are...

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Unformatted text preview: Mathematics 20F Fall 2008 Quiz 1 Name: No aids are allowed. Be sure to write down all row operations used. 1) (5 points) Using elementary row operations, row reduce the following matrix into Reduced Row Echelon Form : 1 2 1 1 2 4 0 −2 . −2 −4 −2 1 Then, answer the following question: if the above matrix is the augmented matrix for some linear system, is the system consistant? 1 2 1 1 1 2 1 1 R ←R2 −2R 2 4 0 −2 −2− − − 1 0 −−−→ 0 −2 −4 −2 −4 −2 1 −2 −4 −2 1 12 1 1 R ←R3 +2R −3− − − 1 0 0 −2 −4 −−−→ 00 0 3 12 1 1 1 R3 ← R3 − − 3− − − → 0 0 −2 −4 00 0 1 12 1 1 R ←R2 +4R −2− − − 3 0 0 −2 0 −−−→ 00 0 1 12 1 0 R ←R1 −R −1− − −3 0 0 −2 0 − − −→ 00 0 1 1210 1 R2 ←− 2 R2 −− −→ 0 0 1 0 −−− 0001 1200 R ←R1 −R −1− − −2 0 0 1 0 − − −→ 0001 If this is the augmented matrix for some linear system, then the last row describes the equation 0 = 1 and hence this system is inconsistent. 2) (5 points) Find the general solution written in parametric vector form to the following matrix equation: 1 3 −5 4 1 4 −8 x = 7 . −3 −7 9 −6 Form the following augmented matrix and row reduce: 1 3 −5 4 1 3 −5 4 R ←R2 −R 1 4 −8 7 −2− − −1 0 1 −3 3 − − −→ −3 −7 9 −6 −3 −7 9 −6 1 3 −5 4 R ←R3 +3R −3− − − 1 0 1 −3 3 −−−→ 0 2 −6 6 1 3 −5 4 R ←R3 −2R −3− − − 2 0 1 −3 3 −−−→ 00 0 0 1 0 4 −5 R ←R1 −3R −1− − − 2 0 1 −3 3 −−−→ 00 0 0 Rewriting the above, we have x1 = −5 − 4x3 x2 = 3 + 3x3 x3 = free or in parametric vector form, −5 −4 x = 3 + x3 3 0 1 ...
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This note was uploaded on 03/17/2011 for the course MATH 20F taught by Professor Buss during the Winter '03 term at UCSD.

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