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Can anyone tell me is it true to say that if I have a matrix P, det(P) is NOT EQUAL to 0, then the vectors that would form the columns of P are linearly independent?

Cheers

- Thread starter sajama
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- #1

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Can anyone tell me is it true to say that if I have a matrix P, det(P) is NOT EQUAL to 0, then the vectors that would form the columns of P are linearly independent?

Cheers

- #2

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Yes because then P is invertible, and thus have maximal rank.

- #3

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Brilliant - cheers! :)

- #4

mathwonk

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recall that any matrix can be rendered into upper diagonal form by repeatedly performing row operations, hence also by repeatedly multiplying by special invertible matrices.

then the determinant is non zero iff the final matrix is actually diagonal and has non zero entries on the diagonal. these can then be made 1's.

hence it has been inverted by matrix multiplication, and the columns are visibly independent, hence were also originally.

this is just one of many ways to see it.

- #5

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Thanks again for the help :)

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