If a,b,c be positive and not all equal, then value of determinant left| {,begin{array}{*{20}{c}}a&am

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

medium

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is

A

$- ve$

B

$=+ ve$

C

Depends on $a,b,c$

D

None of these

(IIT-1982)

Solution

(a) $\Delta = – ({a^3} + {b^3} + {c^3} – 3abc)$

= $ – (a + b + c)\,({a^2} + {b^2} + {c^2} – ab – bc – ca)$

$ = – \frac{1}{2}(a + b + c)\,[{(a – b)^2} + {(b – c)^2} + {(c – a)^2}]$,

which is clearly negative because of the given conditions.

Standard 12

Mathematics

Share

Similar Questions

The number of positive integral solutions $\left| {\,\,\begin{array}{*{20}{c}}{1 – \lambda }&2&1\\{ – 3}&\lambda &{ – 2}\\2&{ – 2}&{1 + \lambda }\end{array}\,\,} \right|$ $= 0$ is

medium

The cubic $\left| {\begin{array}{*{20}{c}}
0&{a – x}&{b – x} \\
{ – a – x}&0&{c – x} \\
{ – b – x}&{ – c – x}&0
\end{array}} \right| = 0$ has a reperated root in $x$ then,

normal

Evaluate the determinants

$\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

easy

The number of values of $\theta \in (0,\pi)$ for which the system of linear equations
$x + 3y + 7z = 0$
$-x + 4y + 7z = 0$
$(sin\,3\theta )x + (cos\,2\theta )y + 2z = 0$ has a non-trivial solution, is

hard

(JEE MAIN-2019)

If ${\left| {\,\begin{array}{{20}{c}}4&1\\2&1\end{array}\,} \right|^2} = \left| {\,\begin{array}{{20}{c}}3&2\\1&x\end{array}\,} \right| – \left| {\,\begin{array}{*{20}{c}}x&3\\{ – 2}&1\end{array}\,} \right|$, then $ x =$

medium