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If $\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]$
is a skew symmetric matrix and $b, c, f$ are non-zero real numbers then $\frac{b}{c}=$
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is a skew symmetric matrix and $b, c, f$ are non-zero real numbers then $\frac{b}{c}=$
Solution:
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Verified Answer
The correct answer is:
$\frac{-d h}{f g}$
$$
\begin{aligned}
& \text { } \mathrm{A}=\left[\begin{array}{lll}
\mathrm{a} & \mathrm{b} & \mathrm{c} \\
\mathrm{d} & \mathrm{e} & \mathrm{f} \\
\mathrm{g} & \mathrm{h} & \mathrm{u}
\end{array}\right] \quad \because \mathrm{A}=-\mathrm{A}^{\mathrm{t}} \\
& \Rightarrow \mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}} \\
& \Rightarrow 2 \mathrm{a}_{\mathrm{ii}}=0 \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \\
& \Rightarrow \mathrm{a}=\mathrm{e}=\mathrm{i}=0
\end{aligned}
$$
$$
\Rightarrow A=\left[\begin{array}{lll}
0 & b & c \\
d & 0 & f \\
g & h & 0
\end{array}\right]
$$
$$
\begin{aligned}
& |\mathrm{A}|=-\mathrm{b}(0-\mathrm{gf})+\mathrm{c}(\mathrm{dh}-0) \\
& \Rightarrow \quad|\mathrm{A}|=\mathrm{bgf}+\mathrm{cdh}
\end{aligned}
$$
$\because \quad$ A is odd order skew symmetric matrix
$$
\Rightarrow|\mathrm{A}|=0
$$
$$
\Rightarrow \mathrm{bgf}+\mathrm{cdh}=0
$$
$\Rightarrow \mathrm{bgf}=-\mathrm{cdh}$
$$
\Rightarrow \frac{\mathrm{b}}{\mathrm{c}}=\frac{-\mathrm{dh}}{\mathrm{fg}}
$$
\begin{aligned}
& \text { } \mathrm{A}=\left[\begin{array}{lll}
\mathrm{a} & \mathrm{b} & \mathrm{c} \\
\mathrm{d} & \mathrm{e} & \mathrm{f} \\
\mathrm{g} & \mathrm{h} & \mathrm{u}
\end{array}\right] \quad \because \mathrm{A}=-\mathrm{A}^{\mathrm{t}} \\
& \Rightarrow \mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}} \\
& \Rightarrow 2 \mathrm{a}_{\mathrm{ii}}=0 \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \\
& \Rightarrow \mathrm{a}=\mathrm{e}=\mathrm{i}=0
\end{aligned}
$$
$$
\Rightarrow A=\left[\begin{array}{lll}
0 & b & c \\
d & 0 & f \\
g & h & 0
\end{array}\right]
$$
$$
\begin{aligned}
& |\mathrm{A}|=-\mathrm{b}(0-\mathrm{gf})+\mathrm{c}(\mathrm{dh}-0) \\
& \Rightarrow \quad|\mathrm{A}|=\mathrm{bgf}+\mathrm{cdh}
\end{aligned}
$$
$\because \quad$ A is odd order skew symmetric matrix
$$
\Rightarrow|\mathrm{A}|=0
$$
$$
\Rightarrow \mathrm{bgf}+\mathrm{cdh}=0
$$
$\Rightarrow \mathrm{bgf}=-\mathrm{cdh}$
$$
\Rightarrow \frac{\mathrm{b}}{\mathrm{c}}=\frac{-\mathrm{dh}}{\mathrm{fg}}
$$
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