Can we split all the columns at once?

Dear Student,

No, we cannot split all coloumns at once.

$$\begin{gathered} \mathrm{Let} ∆ = \left|\begin{array}{ccc}\mathrm{b}+\mathrm{c}& q+r& y+z\\ c+a& \mathrm{r}+\mathrm{p}& z+x\\ a+b& p+q& \mathrm{x}+\mathrm{y}\end{array}\right| \\ \mathrm{Applying} R_1 = R_1+R_2+R_3 \\ ∆ =\left|\begin{array}{ccc}2\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)& 2\left(\mathrm{p}+\mathrm{q}+\mathrm{r}\right)& 2\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)\\ c+a& \mathrm{r}+\mathrm{p}& z+x\\ a+b& p+q& \mathrm{x}+\mathrm{y}\end{array}\right| \\ ∆ = 2\left|\begin{array}{ccc}\mathrm{a}+\mathrm{b}+\mathrm{c}& \mathrm{p}+\mathrm{q}+\mathrm{r}& \mathrm{x}+\mathrm{y}+\mathrm{z}\\ c+a& \mathrm{r}+\mathrm{p}& z+x\\ a+b& p+q& \mathrm{x}+\mathrm{y}\end{array}\right| \\ \mathrm{Applying} R_2 = R_2 - R_1 \mathrm{and} R_3 = R_3-R_1 \\ ∆ =2\left|\begin{array}{ccc}\mathrm{a}+\mathrm{b}+\mathrm{c}& \mathrm{p}+\mathrm{q}+\mathrm{r}& \mathrm{x}+\mathrm{y}+\mathrm{z}\\ -\mathrm{b}& -\mathrm{q}& -y\\ -\mathrm{c}& -\mathrm{r}& -\mathrm{z}\end{array}\right| \\ \mathrm{Applying} R_1 = R_1+R_2+R_3 \\ ∆ = 2\left|\begin{array}{ccc}\mathrm{a}& p& x\\ -b& -\mathrm{q}& -y\\ -\mathrm{c}& -r& -\mathrm{z}\end{array}\right| \\ ∆ = 2{\left(-1\right)}^2\left|\begin{array}{ccc}\mathrm{a}& p& x\\ b& \mathrm{q}& y\\ c& r& \mathrm{z}\end{array}\right| = 2 \left|\begin{array}{ccc}\mathrm{a}& p& x\\ b& \mathrm{q}& y\\ c& r& \mathrm{z}\end{array}\right| \end{gathered}$$



Regards!