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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}\text{a}+\text{b}&2\text{a}+\text{b}&3\text{a}+\text{b}\\2\text{a}+\text{b}&3\text{a}+\text{b}&4\text{a}+\text{b}\\4\text{a}+\text{b}&5\text{a}+\text{b}&6\text{a}+\text{b} \end{vmatrix}$
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Answer

$\begin{vmatrix}\text{a}+\text{b}&2\text{a}+\text{b}&3\text{a}+\text{b}\\2\text{a}+\text{b}&3\text{a}+\text{b}&4\text{a}+\text{b}\\4\text{a}+\text{b}&5\text{a}+\text{b}&6\text{a}+\text{b} \end{vmatrix}$
Apply$: C_3 \rightarrow C_3 - C_2$
$\begin{vmatrix}\text{a}+\text{b}&2\text{a}+\text{b}&\text{a}\\2\text{a}+\text{b}&3\text{a}+\text{b}&\text{a}\\4\text{a}+\text{b}&5\text{a}+\text{b}&\text{a} \end{vmatrix}$
Apply$: C_2 \rightarrow C_2 - C_1​​​​​​​$​​​​​​​
$\begin{vmatrix}\text{a}+\text{b}&\text{a}&\text{a}\\2\text{a}+\text{b}&\text{a}&\text{a}\\4\text{a}+\text{b}&\text{a}&\text{a} \end{vmatrix}$
$=0$
$\because\text{C}_3=\text{C}_2$

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