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Question: Determine whether or not any column in the matrix is a linear combination of other columns. Provide a general method for answering the same question for any n x n matrix A.

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My response:

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Am I am in the right path or completely inaccurate?

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  • $\begingroup$ what you have done is correct. you can also row reduce and see if you get pivots on all three rows. $\endgroup$
    – abel
    Apr 19, 2015 at 20:06
  • $\begingroup$ It should be correct, but if you have for instance a 5x5 matrix, it's boring to calculate the determinant. So, you can check the linear independency of the columns/row by making elementary column/rows operation. I say columns/row because there is a theorem that assures us that the column rank is equal to the row rank, where the rank is just the numer of independent vectors inside the matrix. $\endgroup$ Apr 19, 2015 at 20:07

2 Answers 2

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Almost correct. You say:

If $|A| = 0$, then each column is a linear combination of the other.

You should say:

If $|A| = 0$, then some column is a linear combination of some others.

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This is good. When at least, one row or column is linear dependent, the matrix determinant is zero. In other words, when dimension is greater than the range of the matrix.

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