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Determinants (p-1) — Maths (commerce) STD 11 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 11 Commerce / ArtsMaths (commerce)Determinants (p-1)1 Mark
Question
Without expanding, evaluate the following determinants.

(i) $\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{array}\right|$

✓

Answer

Let $D=\left|\begin{array}{lll}1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b\end{array}\right|$
Applying $\mathrm{C}_3 \rightarrow \mathrm{C}_3+\mathrm{C}_2$, we get
$D=\left|\begin{array}{lll}1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c\end{array}\right|$
Taking $(\mathrm{a}+\mathrm{b}+\mathrm{c})$ common from $\mathrm{C}_3$, we get
$
\begin{aligned}
& D=(a+b+c)\left|\begin{array}{lll}
1 & a & 1 \\
1 & b & 1 \\
1 & c & 1
\end{array}\right| \\
& \therefore \quad \mathrm{D}=(\mathrm{a}+\mathrm{b}+\mathrm{c})(0) \\
& \therefore \quad \mathrm{D}=0 \\
&
\end{aligned}
$
$\ldots\left[\because \mathrm{C}_1\right.$ and $\mathrm{C}_3$ are identical $]$

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