First slide

Introduction to Determinants

Question

a₁b₁c₁, a₂b₂c₂, and a₃b₃c₃ are three-digit even natural numbers and $Δ = |\begin{array}{lll} c_{1} a_{1} b_{1} \\ c_{2} a_{2} b_{2} \\ c_{3} a_{3} b_{3} \end{array}|$ , then $∆$ =

Difficult

A

divisible by 2 but not necessarily by 4
B

divisible by 4 but not necessarily by 8
C

divisible by 8
D

none of these

Solution

As a₁b₁c₁, a₂b₂c₂, and a₃b₃c₃ are even natural numbers, each of c₁, c₂, c₃ is divisible by 2. Let $c_{i} = 2 k_{i}$ for i = 1, 2, 3. Thus,
$Δ = 2 |\begin{array}{lll} k_{1} a_{1} b_{1} \\ k_{2} a_{2} b_{2} \\ k_{3} a_{3} b_{3} \end{array}| = 2 m$
where m is some natural number. Thus, $∆$ is divisible by 2.That $∆$ may not be divisible by 4 can be seen by taking the three numbers as 112, 122, and 134. Note that
$Δ = |\begin{array}{lll} 2 1 1 \\ 2 1 2 \\ 4 1 3 \end{array}| = 2$
which is divisible by 2 but not by 4.

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