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# 0 is indeﬁnite because h 2 8 e 1 0 thus 1 0 and 1 0

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0) is indeﬁnite because H 2 = - 8 e - 1 < 0. Thus (1 , 0) and ( - 1 , 0) are saddlepoints. The function has no maximum. 5. Consider the matrix A = 3 2 1 3 1 0 3 3 1 a ) What is the rank of A ? Answer: The rank of A is 3. One way to see this is to compute the determinant. It is (3)(1)(1) + (3)(3)(1) + (3)(2)(0) - (3)(1)(1) - (3)(2)(1) - (3)(3)(0) = 3 + 9 - 3 - 6 = 3. Since it is non-zero, the matrix is invertible. As it has 3 rows, it has rank 3. b ) How many linearly independent rows does A have? Columns? Answer: The row rank is the maximum number of linearly independent rows, the column rank is the maximum number of linearly independent columns. Both must be equal to the rank of the matrix. Since the rank is 3, there are 3 linearly independent rows, and 3 linearly independent columns. c ) How many solutions are there to the equation A x = 3 2 1 ? Answer: In part (a) we found that A is invertible. It follows that the system of equations has exactly one solution. For the record, the solution is (4 / 3 , - 2 , 3) t .
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