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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Evaluate the following:
$\begin{vmatrix}\text{a}+\text{x}&\text{y}&\text{z}\\\text{x}&\text{a}+\text{y}&\text{z}\\\text{x}&\text{y}&\text{a}+\text{z}\end{vmatrix}$
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Answer

Let $\triangle=\begin{vmatrix}\text{a}+\text{x}&\text{y}&\text{z}\\\text{x}&\text{a}+\text{y}&\text{z}\\\text{x}&\text{y}&\text{a}+\text{z}\end{vmatrix}$
Applying $R_1 \rightarrow R_1 - R_2$ and $R_3 \rightarrow R_3 - R_2$
$\triangle=\begin{vmatrix}\text{a}&-\text{a}&0\\\text{x}&\text{a}+\text{y}&\text{z}\\0&-\text{a}&\text{a}\end{vmatrix}$
Applying $C_2 \rightarrow C_2 + C_1​​​​​​​$
$\triangle=\begin{vmatrix}\text{a}&0&0\\\text{x}&\text{a}+\text{y}&\text{z}\\0&-\text{a}&\text{a}\end{vmatrix}$
$\triangle=\text{a}[\text{a}(\text{a}+\text{x}+\text{y})+\text{az}]+0+0$
$\triangle=\text{a}^2(\text{a}+\text{x}+\text{y}+\text{z})$

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