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3 and 4 .Determinants and Matrices
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Let $a, b, c > 0$ and $\Delta = \left| \begin{gathered}
a + b\,\,b\,\,c \hfill \\
b\, + \,c\,\,c\,\,\,a \hfill \\
c + a\,\,a\,\,b \hfill \\
\end{gathered} \right| ,$ then which of the following is not correct?
A
$\Delta = -[a^3 + b^3 + c^3 - 3abc$]
B
$\Delta\leq 0$
C
$\Delta = 0 \Rightarrow\ a + b + c = 0$
D
$\Delta = 0$ if $a = b = c$
Solution
$c_{1} \rightarrow c_{1}-c_{2}$
$\Delta=\left|\begin{array}{lll}{a} & {b} & {c} \\ {b} & {c} & {a} \\ {c} & {a} & {b}\end{array}\right|=-\left[a^{3}+b^{3}+c^{3}-3 a b c\right]$
Or $\Delta=-\frac{1}{2}(a+b+c)\left[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}\right]$
So $\Delta<0$
Also $\Delta=0 \Rightarrow \mathrm{a}=\mathrm{b}=\mathrm{c}(\therefore \mathrm{a}+\mathrm{b}+\mathrm{c}>0)$
Since $a+b+c \neq 0$
Standard 12
Mathematics
