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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
Question
State True or False for the statements of the following Exercise:
Let $ \begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16,$ then $\Delta_1=\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b} +\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+ \text{z}&\text{c}+\text{r}\end{vmatrix}=32.$
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Answer

True.
Solution:
Given $ \begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16$
Now $\Delta_1=\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b}+\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+\text{z}&\text{c}+\text{r}\end{vmatrix}=32$
$\big[\text{Applying C}_1\rightarrow\text{C}_1+\text{C}_2+\text{C}_3\big]$
$=\begin{vmatrix}2(\text{p}+\text{x}+\text{a})&\text{a}+\text{x}&\text{a}+\text{p}\\2(\text{q}+\text{y}+\text{b})&\text{b} +\text{y}&\text{b}+\text{q}\\2(\text{r}+\text{z}+\text{c})&\text{c}+\text{z}&\text{c}+\text{r}\end{vmatrix}$
$\big[\text{Applying C}_1\rightarrow\text{C}_1-\text{C}_2\big]$
$=2\begin{vmatrix}\text{p}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}&\text{b}+\text{y}&\text{b}+\text{q}\\\text{r}&\text{c}+\text{z}&\text{c}+\text{r}\end{vmatrix}$
$\big[\text{Applying C}_3\rightarrow\text{C}_3-\text{C}_1\big]$
$=2\begin{vmatrix}\text{p}&\text{a}+\text{x}&\text{a}\\\text{q}&\text{b}+\text{y}&\text{b}\\\text{r}&\text{c}+\text{z}&\text{c}\end{vmatrix}$
$\big[\text{Applying C}_2\rightarrow\text{C}_2-\text{C}_3\big]$
$=2\begin{vmatrix}\text{p}&\text{x}&\text{a}\\\text{q}&\text{y}&\text{b}\\\text{r}&\text{z}&\text{c}\end{vmatrix}=2\begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=2\times16=32$

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