On a Property of Related Determinants
Abstract Generated abstract
This note studies four “related” third order determinants formed cyclically from the columns of a three by four matrix. It proves that if two such determinants vanish and at least one of their common second order minors is nonzero, then the other two related determinants must also vanish, using expansions by columns and the resulting homogeneous linear relations among the minors. It also treats the complementary case: if two related determinants vanish while a third is nonzero, then the common minors of the first two vanish, which implies proportionality of the corresponding column elements.
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V. L. Nekrasov.
On a Property of Related Determinants.
The determinants
\[ \Delta_0= \begin{vmatrix} a_1&b_1&c_1\\ a_2&b_2&c_2\\ a_3&b_3&c_3 \end{vmatrix},\quad \Delta_1= \begin{vmatrix} b_1&c_1&d_1\\ b_2&c_2&d_2\\ b_3&c_3&d_3 \end{vmatrix},\quad \Delta_2= \begin{vmatrix} c_1&d_1&a_1\\ c_2&d_2&a_2\\ c_3&d_3&a_3 \end{vmatrix},\quad \Delta_3= \begin{vmatrix} d_1&a_1&b_1\\ d_2&a_2&b_2\\ d_3&a_3&b_3 \end{vmatrix}, \tag{1} \]
composed of the elements of the matrix
\[ \begin{Vmatrix} a_1&b_1&c_1&d_1\\ a_2&b_2&c_2&d_2\\ a_3&b_3&c_3&d_3 \end{Vmatrix} \]
in cyclic order, we shall call related.
1. Let two of the determinants (1), say \(\Delta_0\) and \(\Delta_3\), be equal to \(0\), and let at least one of their common minors
\[ \begin{vmatrix} a_1&b_1\\ a_2&b_2 \end{vmatrix},\quad \begin{vmatrix} a_2&b_2\\ a_3&b_3 \end{vmatrix},\quad \begin{vmatrix} a_3&b_3\\ a_1&b_1 \end{vmatrix} \tag{2} \]
be different from \(0\).
Expanding \(\Delta_0\) by the elements of the third column, and \(\Delta_3\) by those of the first column, we obtain
\[ c_1 \begin{vmatrix} a_2&b_2\\ a_3&b_3 \end{vmatrix} +c_2 \begin{vmatrix} a_3&b_3\\ a_1&b_1 \end{vmatrix} +c_3 \begin{vmatrix} a_1&b_1\\ a_2&b_2 \end{vmatrix} =0, \tag{3} \]
\[ d_1 \begin{vmatrix} a_2&b_2\\ a_3&b_3 \end{vmatrix} +d_2 \begin{vmatrix} a_3&b_3\\ a_1&b_1 \end{vmatrix} +d_3 \begin{vmatrix} a_1&b_1\\ a_2&b_2 \end{vmatrix} =0; \]
besides, as is known,
\[ a_1 \begin{vmatrix} a_2&b_2\\ a_3&b_3 \end{vmatrix} +a_2 \begin{vmatrix} a_3&b_3\\ a_1&b_1 \end{vmatrix} +a_3 \begin{vmatrix} a_1&b_1\\ a_2&b_2 \end{vmatrix} =0, \tag{4} \]
\[ b_1 \begin{vmatrix} a_2&b_2\\ a_3&b_3 \end{vmatrix} +b_2 \begin{vmatrix} a_3&b_3\\ a_1&b_1 \end{vmatrix} +b_3 \begin{vmatrix} a_1&b_1\\ a_2&b_2 \end{vmatrix} =0. \tag{5} \]
Since the minors (2)—by hypothesis—are not all equal to \(0\), and since they satisfy, first, the three homogeneous equations (3) and (4), and, second, (5) and (3), the corresponding determinants of the coefficients of these equations must vanish; and these determinants are \(\Delta_2\) and \(\Delta_1\); thus,
If, of four related determinants, two vanish, and at least one of their common minors is different from \(0\), then the remaining two determinants are equal to zero as well.
2. Let now \(\Delta_0\) and \(\Delta_3\) be equal to \(0\), and let at least one of the other two determinants, for example \(\Delta_2\), be different from \(0\).
In this case it follows from (3) and (4) that the minors (2) must vanish; from their equality to \(0\) it follows that
\[ \frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}, \]
therefore:
If two related determinants are equal to \(0\), and the third is different from \(0\), then the common minors of the first two determinants are equal to \(0\), and the elements entering into them are proportional.