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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
Without expanding, prove that:
$\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\\\text{a}&\text{b}&\text{c}\end{vmatrix}=\begin{vmatrix}\text{y}&\text{b}&\text{q}\\\text{x}&\text{a}&\text{p}\\\text{z}&\text{c}&\text{r}\end{vmatrix}$
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Answer

$=\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\\\text{a}&\text{b}&\text{c}\end{vmatrix}$ $\text{R}_2\leftrightarrow\text{R}_3$
$=-\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{a}&\text{b}&\text{c}\\\text{p}&\text{q}&\text{r}\end{vmatrix}$ $\text{R}_1\leftrightarrow\text{R}_2$
$=\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}$
$\begin{vmatrix}\text{y}&\text{b}&\text{q}\\\text{x}&\text{a}&\text{p}\\\text{z}&\text{c}&\text{r}\end{vmatrix}$
$=\begin{vmatrix}\text{y}&\text{x}&\text{z}\\\text{b}&\text{a}&\text{c}\\\text{q}&\text{p}&\text{r}\end{vmatrix}$ $\text{C}_1\leftrightarrow\text{C}_2$
$=-\begin{vmatrix}\text{x}&\text{y}&\text{z}\\\text{a}&\text{b}&\text{c}\\\text{p}&\text{q}&\text{r}\end{vmatrix}$ $\text{R}_1\leftrightarrow\text{R}_2$
$=\begin{vmatrix}\text{a}&\text{b}&\text{c}\\\text{x}&\text{y}&\text{z}\\\text{p}&\text{q}&\text{r}\end{vmatrix}$

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