Let D1, D2, D3 ldots Dn be the set of third order determinants that can be made with the distinct non-zero real numbers a1, a2, ldots, a9, then

Question

Let D1, D2, D3 ldots Dn be the set of third order determinants that can be made with the distinct non-zero real numbers a1, a2, ldots, a9, then (A)  displaystyle∑i=1n Di=1 (B)  displaystyle∑i=1n Di=0 (C) D i=Di, ∀i, j (D) None of these

Tardigrade Admin · Accepted Answer

Total number of third order determinants is

9!

As the number of determinants are even and in which there are

\frac{9 !}{2}

pairs of determinants which are obtained by changing two consecutive rows
So,

i = 1 \sum n D_{i} = 0

Q. Let ${D_{1}, D_{2}, D_{3} \dots D_{n}}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_{1}, a_{2}, \dots, a_{9}$ , then

$i = 1 \sum n D_{i} = 1$

$i = 1 \sum n D_{i} = 0$

$D i = D_{i}, \forall_{i}, j$

None of these

Total number of third order determinants is $9!$
As the number of determinants are even and in which there are $\frac{9 !}{2}$ pairs of determinants which are obtained by changing two consecutive rows
So, $i = 1 \sum n D_{i} = 0$

Q. Let {D1​,D2​,D3​…Dn​} be the set of third order determinants that can be made with the distinct non-zero real numbers a1​,a2​,…,a9​, then

i=1∑n​Di​=1

i=1∑n​Di​=0

Di=Di​,∀i​,j