Prove, without expanding, that \begin{vmatrix} 1 &a &a^2-bc \\ 1 &b &b^2-ca \\ 1 &c &c^2-ab \end{vmatrix} vanishes.
Any hints ?
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$\begingroup$ what do you mean with "without expanding"? $\endgroup$– Bman72May 23, 2014 at 15:25
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$\begingroup$ not trying to explicitly calculate the determinant (Laplace's expansion) $\endgroup$– square_oneMay 23, 2014 at 15:27
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2$\begingroup$ You can give a nontrivial linear combination of the columns (or rows) that equals $0$ to show that it vanishes. $\endgroup$– ServaesMay 23, 2014 at 15:29
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5 Answers
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Let
$$f(a)=\begin{vmatrix} 1 &a &a^2-bc \\ 1 &b &b^2-ca \\ 1 &c &c^2-ab \end{vmatrix}$$ then it's easy to see that $f$ is a polynomial on $a$ with degree at most $2$ and $f(b)=f(c)=0$ so $$f(a)=\lambda(a-b)(a-c)$$ now, WLOG assume that $bc\ne0$, take $a=0$; we see easily that $f(0)=0$ hence $\lambda=0$.
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\begin{align} \begin{vmatrix} 1 & a & a^2-bc \\ 1 & b & b^2-ca \\ 1 & c & c^2-ab \end{vmatrix} &=\begin{vmatrix} 1 & a & a^2-bc \\ 0 & b-a & b^2-a^2+bc-ca \\ 0 & c-b & c^2-b^2+ca-ab \end{vmatrix} \\ &=\begin{vmatrix} b-a & b^2-a^2+bc-ca \\ c-b & c^2-b^2+ca-ab \end{vmatrix} \\ &=\begin{vmatrix} b-a & (b-a)(b+a+c) \\ c-b & (c-b)(c+b+a) \end{vmatrix} \end{align}
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1$\begingroup$ This is very nice to my honest opinion, the only thing is that the OP is not allowed to use the laplace expansion, but it is enough to leave the first row and column that you correctly canceled because it contained the column $\begin{pmatrix}1 \\0\\0\end{pmatrix}$ $\endgroup$– Bman72May 23, 2014 at 15:37
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$\begingroup$ What does "OP" mean... ? I have seen this elsewhere as well ! $\endgroup$ May 23, 2014 at 15:40
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6$\begingroup$ @user148176 the OP is you! the Original Poster $\endgroup$– Bman72May 23, 2014 at 15:43
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$\begingroup$ @user148176 Also means 'Original Post' - i.e. the thing or the person behind it, obvious from context. (And it's hard to imagine that the difference even matters, other than for correct grammar). $\endgroup$– OJFordMay 24, 2014 at 2:23
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1$\begingroup$ @Ale "We have met the OP and he is us." -Walt Kelly $\endgroup$ May 24, 2014 at 18:14
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e.g. if your matrix is $A$, consider $(b-c,c-a,a-b) A$
answered May 23, 2014 at 15:29
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Add $(ab+bc+ca)$ times the first column to the last and find the common factor of the last column.
answered May 23, 2014 at 15:31
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Simply add a multiple of the first column to the last $$\begin{vmatrix} 1 & a & a^2-bc\\1&b&b^2-ac\\1&c&c^2-ab\end{vmatrix} =\begin{vmatrix} 1 & a & a^2-bc+(1)(ab+ac+bc)\\1&b&b^2-ac+(1)(ab+ac+bc)\\1&c&c^2-ab+(1)(ab+ac+bc)\end{vmatrix} = \begin{vmatrix} 1 & a & a(a+b+c)\\1&b&b(a+b+c)\\1&c&c(a+b+c)\end{vmatrix} = 0$$ The final step follows because the second and third column are linearly dependent.
answered May 24, 2014 at 17:10
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$\begingroup$ Just realized this is what LutzL said to do $\endgroup$ May 24, 2014 at 17:10