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I've just learnt that det(A) = 0 when the columns of a matrix are linearly dependent, but what does that mean? Could you give me an easy to follow example with numbers please?

Thank you!

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  • $\begingroup$ The identity matrix has determinant $1$. Linear independence of the columns means that the column vectors of the matrix are linearly independent. You should review this definition (e.g. google "linear independence"). $\endgroup$
    – Dave
    Mar 30, 2018 at 23:01
  • $\begingroup$ Oh sorry, honest mistake. Ok thanks $\endgroup$
    – Jess
    Mar 30, 2018 at 23:01
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    $\begingroup$ You mean the determinant is $0$ when the column are linearly dependent that is not linearly independent. $\endgroup$
    – saulspatz
    Mar 30, 2018 at 23:17
  • $\begingroup$ @Jess Please remember that you can choose an answer among the given if the OP is solved, more details here meta.stackexchange.com/questions/5234/… $\endgroup$
    – user
    Apr 1, 2018 at 8:53

2 Answers 2

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By definition $\{\vec v_i\}$ are linearly independent if

$$\sum a_i \vec v_i=0 \iff a_i=0 \quad\forall i$$

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The matrix $\begin{bmatrix} 1 & 0 & 5\\ 2 & 4 & 10\\ -5 & 0 & -25 \end{bmatrix}$ has linearly DEPENDENT columns because the last column can be obtained from the first one only by multiplying by $5$. In this case, the determinant of the matrix is $0$ and so the matrix is not invertible.

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    $\begingroup$ Thanks! This is the kind of answer i was looking for $\endgroup$
    – Jess
    Mar 31, 2018 at 10:20
  • $\begingroup$ @Jess Glad you understood! $\endgroup$
    – mandella
    Mar 31, 2018 at 10:22

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