Without expanding determinants, find the value of (i) | lll 10 & … - Vidyadip

Question

Without expanding determinants, find the value of

(i) $\left|\begin{array}{lll}10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153\end{array}\right|$

✓

Answer

Let $D=\left|\begin{array}{lll}10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153\end{array}\right|$
Applying $\mathrm{C}_3 \rightarrow \mathrm{C}_3-\mathrm{C}_2$, we get
$
\mathrm{D}=\left|\begin{array}{lll}
10 & 57 & 50 \\
12 & 64 & 60 \\
15 & 78 & 75
\end{array}\right|
$
Taking (5) common from $\mathrm{C}_3$, we get
$
\begin{aligned}
& \mathrm{D}=5\left|\begin{array}{lll}
10 & 57 & 10 \\
12 & 64 & 12 \\
15 & 78 & 15
\end{array}\right| \\
& =5(0) \quad \ldots\left[\because C_1 \text { and } \mathrm{C}_3 \text { are identical }\right] \\
& =0 \quad \text { } \\
&
\end{aligned}
$

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