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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants3 Marks
Question
Write the value of the determinant $\begin{vmatrix}\text{x}+\text{y}&\text{y}+\text{z}&\text{z}+\text{x}\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$
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Answer

$\begin{vmatrix}\text{x}+\text{y}&\text{y}+\text{z}&\text{z}+\text{x}\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$
$=\begin{vmatrix}\text{x}+\text{y}+\text{z}&\text{x}+\text{y}+\text{z}&\text{z}+\text{x}+\text{y}\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$ [Applying $R_1 → R_1 + R_2$]
$=(\text{x}+\text{y}+\text{z})\begin{vmatrix}1&1&1\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$ [Taking (x + y + z) common from $R_1$]
$=(\text{x}+\text{y}+\text{z})\begin{vmatrix}1&1&1\\\text{z}&\text{x}&\text{y}\\0&0&0 \end{vmatrix}$
$=0$ [Expanding along the last row]
Hence, the value of the determinant $\begin{vmatrix}\text{x}+\text{y}&\text{y}+\text{z}&\text{z}+\text{x}\\\text{z}&\text{x}&\text{y}\\-3&-3&-3 \end{vmatrix}$ is 0

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