I am currently doing a practice final for a Linear Algebra Class.

In it I am given the following statement and asked to determine whether it is true or false.

"If det(A) = 0, then two rows or two columns of A are the same, or a row or a column of A is zero."

Obviously if a row or column of A is zero then when transforming that matrix to a diagonal matrix it will have a zero in that row or column; this in turn will cause the determinant (the product of diagonal elements) to be zero.

Similarly, if two rows of the matrix are the same then when performing row operations to make it a diagonal matrix will create a row of zeros which cause the determinant to be zero.

The answer provided to me is that this statement is false, so this would mean that the statement concerning the identical columns is false.

However, if vectors are a linear combination of each other (obviously identical vectors are), then when reducing a matrix to reduced echelon form it will have less pivot columns than it does rows. Again this would imply that there is a zero element on the diagonal and the determinant would be zero.

I want to be certain without a doubt that the answer provided to me is false.

Thank you for your help and time.

  • 2
    $\begingroup$ Careful -- the question in your title is different from the question that you've written in the body of the text. The statement in the title is true, the one in the text is false. $\endgroup$ – Cam McLeman Aug 22 '15 at 0:37
  • $\begingroup$ If $\det(A)=0$, then A could have two columns which are scalar multiples of each other. This is a counter-example to the statement, which leads us to conclude that it is false. $\endgroup$ – eloiprime Aug 22 '15 at 0:46

The answer provided to you is true. The statement is false.

Your problem is that you are looking at it backwards. You are noting that either of the two conditions in the "then" clause will cause the determinant to be 0. But that is not what the statement is saying. The question you need to ask is "are these the only ways a determinant can be zero? Or are there other conditions that would lead to a zero determinant?


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