quiz1-3

# quiz1-3 - A is at least 2 However A has only two columns so...

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Quiz 1 Name: Math 240 - Calculus III January 27, 2009 Note: In order to receive full credit, you must show work that justiﬁes your answer. 1. Solve the following system: a + b + c = 5 a + 2 b + 3 c = 12 2 a - b - c = 1 Solution : We form the augmented matrix and row-reduce 1 1 1 5 1 2 3 12 2 - 1 - 1 1 1 1 1 5 0 1 2 7 0 - 3 - 3 - 9 1 0 - 1 - 2 0 1 2 7 0 0 3 12 1 0 0 2 0 1 0 - 1 0 0 1 4 Thus, the solution is a = 2, b = - 1, and c = 4. Substituting these values back into the original equations, we see that our answer is correct. 2. Find the rank of the matrix A = 1 0 1 0 2 0 3 0 3 . Solution : The row-reduced form of A has two nonzero rows: 1 0 1 0 2 0 3 0 3 1 0 1 0 2 0 0 0 0 Thus, the rank of A is 2.

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Quiz 1 Name: Math 240 - Calculus III January 29, 2009 Note: In order to receive full credit, you must show work that justiﬁes your answer. 1. Find the rank of the matrix A = 1 2 3 4 5 6 7 8 . Solution : The ﬁrst two rows of A are linearly independent (why?), so the rank of
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Unformatted text preview: A is at least 2. However, A has only two columns, so its rank is at most 2. Therefore, the rank of A is 2. Alternately, we could row-reduce A and see that the reduced form has 2 nonzero rows. 2. Find a number n such that the vectors h 1 , 2 , i , h , 2 , 3 i , and h 3 , 2 , n i are linearly dependent . Solution : Consider the matrix whose rows are the given vectors: 1 2 0 0 2 3 3 2 n . The row vectors are linearly dependent if and only if we can row reduce and obtain at least one row of zeros. Row-reducing: 1 2 0 0 2 3 3 2 n → 1 2 2 3-4 n → 1 2 0 2 3 0 0 n + 6 The reduced form of the matrix has a row of zeros if and only if n =-6. Therefore, the given vectors are linearly dependent when n =-6....
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quiz1-3 - A is at least 2 However A has only two columns so...

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