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6 Nov 2019
This problem shows why we should always be careful when doing mathematics on a computer. Here was an odd problem I encountered when making written assignment 5. I defined the following matrix in maple: A = [-2 1 0 4 5 -3 -2 1 2 x 9 1 1 -1 -2 9 -18 10 4 14]. The question was supposed to be "Determine all values of x such that the rank of A is equal to 2". Running the GaussianElimination command in Maple on this matrix produced [-2 0 0 0 5 -1/2 0 0 2 x + 1 5 - 4x 0 1 -1/2 0 0 -18 1 0 0]. Now to make the rank of A equal to 2, we need to have exactly two pivot columns in this echelon form. This would seem to imply that we have to make 5 - 4x = 0 which would give a value of x = 5/4. However, if you substitute this into the matrix and row reduce to an echelon form, you get [-2 0 0 0 5 -1/2 0 0 2 9/4 109/2 0 1 -1/2 0 0 -18 1 0 0] which clearly has three pivot columns, so that the rank of A would actually be three. Furthermore, here is another curiosity. If we swap rows 1 and 4 in the original matrix A and run the GaussianElimination command, we get the matrix [4 0 0 0 1 -13/4 0 0 1 x - 1/4 119/13 - 8/13x 0 9 -13/4 0 0 14 13/2 0 0] which implies that for the rank of A to be 2, x must be equal to 119/8, which is preposterous as 119/8 is definitely not equal to 5/4. The question to you is this: Why did this method not work and what did I do wrong? In order to get credit for this question you must give me a detailed explanation about what happened in the calculation. Once you have figured this out, producing an example of your own illustrating the error will be required. Show transcribed image text
This problem shows why we should always be careful when doing mathematics on a computer. Here was an odd problem I encountered when making written assignment 5. I defined the following matrix in maple: A = [-2 1 0 4 5 -3 -2 1 2 x 9 1 1 -1 -2 9 -18 10 4 14]. The question was supposed to be "Determine all values of x such that the rank of A is equal to 2". Running the GaussianElimination command in Maple on this matrix produced [-2 0 0 0 5 -1/2 0 0 2 x + 1 5 - 4x 0 1 -1/2 0 0 -18 1 0 0]. Now to make the rank of A equal to 2, we need to have exactly two pivot columns in this echelon form. This would seem to imply that we have to make 5 - 4x = 0 which would give a value of x = 5/4. However, if you substitute this into the matrix and row reduce to an echelon form, you get [-2 0 0 0 5 -1/2 0 0 2 9/4 109/2 0 1 -1/2 0 0 -18 1 0 0] which clearly has three pivot columns, so that the rank of A would actually be three. Furthermore, here is another curiosity. If we swap rows 1 and 4 in the original matrix A and run the GaussianElimination command, we get the matrix [4 0 0 0 1 -13/4 0 0 1 x - 1/4 119/13 - 8/13x 0 9 -13/4 0 0 14 13/2 0 0] which implies that for the rank of A to be 2, x must be equal to 119/8, which is preposterous as 119/8 is definitely not equal to 5/4. The question to you is this: Why did this method not work and what did I do wrong? In order to get credit for this question you must give me a detailed explanation about what happened in the calculation. Once you have figured this out, producing an example of your own illustrating the error will be required.
Show transcribed image text Irving HeathcoteLv2
2 Aug 2019