If $\mathrm{a, b, c}$ are positive and unequal, show that value of the determinant $\Delta=\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is negative.
If $\mathrm{a, b, c}$ are positive and unequal, show that value of the determinant $\Delta=\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is negative.
Applying $\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}$ to the given determinant, we get
$\Delta = \left| {\begin{array}{*{20}{l}}
{a + b + c}&b&c \\
{a + b + c}&c&a \\
{a + b + c}&a&b
\end{array}} \right| = (a + b + c)\left| {\begin{array}{*{20}{c}}
1&b&c \\
1&c&a \\
1&a&b
\end{array}} \right|$
${ = (a + b + c)\left| {\begin{array}{*{20}{c}}
1&b&c \\
0&{c - b}&{a - c} \\
0&{a - b}&{b - c}
\end{array}} \right|}$ ${{\text{(Applying }}{{\text{R}}_2} \to {{\text{R}}_2} - {{\text{R}}_1},{\text{ and }}{{\text{R}}_3} \to {{\text{R}}_3} - {{\text{R}}_1})}$
$ = (a + b + c)[(c - b)(b - c) - (a - c)(a - b)]$ ${\text{(Expanding along }}{{\text{C}}_1}{\text{ ) }}$
$ = (a + b + c)\left( { - {a^2} - {b^2} - {c^2} + ab + bc + ca} \right)$
which is negative (since $\left.a+b+c>0 \text { and }(a-b)^{2}+(b-c)^{2}+(c-a)^{2}>0\right)$
Similar Questions
$$f(x)=\left| {\begin{array}{*{20}{c}} {{{\sin }^2}x}&{ - 2 + {{\cos }^2}x}&{\cos 2x} \\ {2 + {{\sin }^2}x}&{{{\cos }^2}x}&{\cos 2x} \\ {{{\sin }^2}x}&{{{\cos }^2}x}&{1 + \cos 2x} \end{array}} \right| ,x \in[0, \pi]$$
Then the maximum value of $f(x)$ is equal to $.....$
$$f(x)=\left| {\begin{array}{*{20}{c}} {{{\sin }^2}x}&{ - 2 + {{\cos }^2}x}&{\cos 2x} \\ {2 + {{\sin }^2}x}&{{{\cos }^2}x}&{\cos 2x} \\ {{{\sin }^2}x}&{{{\cos }^2}x}&{1 + \cos 2x} \end{array}} \right| ,x \in[0, \pi]$$
Then the maximum value of $f(x)$ is equal to $.....$
- [JEE MAIN 2021]
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ is not depend
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ is not depend
Value of $\left| {\begin{array}{*{20}{c}}
{{{(b + c)}^2}}&{{a^2}}&{{a^2}} \\
{{b^2}}&{{{(a + c)}^2}}&{{b^2}} \\
{{c^2}}&{{c^2}}&{{{(a + b)}^2}}
\end{array}} \right|$ is equal to
Value of $\left| {\begin{array}{*{20}{c}}
{{{(b + c)}^2}}&{{a^2}}&{{a^2}} \\
{{b^2}}&{{{(a + c)}^2}}&{{b^2}} \\
{{c^2}}&{{c^2}}&{{{(a + b)}^2}}
\end{array}} \right|$ is equal to
If $a, b, c$ are all different from zero and $\left| {\begin{array}{*{20}{c}} {1 + a}&1&1\\ 1&{1 + b}&1\\ 1&1&{1 + c} \end{array}} \right| = 0$ , then the value of $a^{-1} + b^{-1} + c^{-1}$ is
If $a, b, c$ are all different from zero and $\left| {\begin{array}{*{20}{c}} {1 + a}&1&1\\ 1&{1 + b}&1\\ 1&1&{1 + c} \end{array}} \right| = 0$ , then the value of $a^{-1} + b^{-1} + c^{-1}$ is
By using properties of determinants, show that:
$\left|\begin{array}{ccc}0 & a & -b \\ -a & 0 & -c \\ b & c & 0\end{array}\right|=0$
By using properties of determinants, show that:
$\left|\begin{array}{ccc}0 & a & -b \\ -a & 0 & -c \\ b & c & 0\end{array}\right|=0$