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  • State True or False for the statements If the determinant xplusa pplusu 1plusf splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8

State True or False for the statements

If the determinant \left|\begin{array}{ccc} x+a & p+u & 1+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h \end{array}\right|  splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8

Answers (1)

\\ Given \left|\begin{array}{ccc}x+a & p+u & 1+f \\ y+b & q+v & m+g \\ z+c & r+w & n+h\end{array}\right|$ \\Split row 1\\ $\Rightarrow\left|\begin{array}{ccc}\mathrm{x}+\mathrm{a} & \mathrm{p}+\mathrm{u} & 1+\mathrm{f} \\ \mathrm{y}+\mathrm{b} & \mathrm{q}+\mathrm{v} & \mathrm{m}+\mathrm{g} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h}\end{array}\right|=\left|\begin{array}{ccc}\mathrm{x} & \mathrm{p} & 1 \\ \mathrm{y}+\mathrm{b} & \mathrm{q}+\mathrm{v} & \mathrm{m}+\mathrm{g} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h}\end{array}\right|+\left|\begin{array}{ccc}\mathrm{a} & \mathrm{u} & \mathrm{f} \\ \mathrm{y}+\mathrm{b} & \mathrm{q}+\mathrm{v} & \mathrm{m}+\mathrm{g} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h}\end{array}\right|$

Split row 2

\\ \begin{array}{ccc} \left|\begin{array}{ccc} \mathrm{x} & \mathrm{p} & 1 \\ \mathrm{y} & \mathrm{q} & \mathrm{m} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h} \end{array}\right|+\left|\begin{array}{ccc} \mathrm{a} & \mathrm{u} & \mathrm{f} \\ \mathrm{y} & \mathrm{q} & \mathrm{m} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h} \end{array}\right|+\left|\begin{array}{ccc} \mathrm{x} & \mathrm{p} & \mathrm{l} \\ \mathrm{b} & \mathrm{v} & \mathrm{g} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h} \end{array}\right| \\ & +\left|\begin{array}{ccc} \mathrm{a} & \mathrm{u} & \mathrm{f} \\ \mathrm{b} & \mathrm{v} & \mathrm{g} \\ \mathrm{z}+\mathrm{c} & \mathrm{r}+\mathrm{w} & \mathrm{n}+\mathrm{h} \end{array}\right| \end{array}

We can split all the rows in the same way. Thus the statement given in the question is true.

 

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infoexpert22

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