logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Model Papers
0 votes

prove that : $ \begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc \bigg( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1 \bigg) = (bc+ca+ab+abc).$

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • Elementary transformation can be done by
  • (i) Interchanging any two rows or column.
  • (ii) The addition of elements of any row or column.
  • If $A=\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}$
  • Then $|A|=a_{11}(a_{22}\times a_{33}-a_{23}\times a_{32})-a_{12}(a_{21}\times a_{33}-a_{23}\times a_{31})+a_{13}(a_{21}\times a_{32}-a_{22}\times a_{31})$$
Let $\Delta=\begin{vmatrix}1+a & 1 & 1\\1 & 1+b & 1\\1 & 1 &1+c\end{vmatrix}$
Take out the factors a,b,c from $R_1,R_2$ and $R_3$
$\Delta=abc\begin{vmatrix}\frac{1}{a}+1 & \frac{1}{a} & \frac{1}{a}\\\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b}\\\frac{1}{c} & \frac{1}{c} &\frac{1}{c}+1\end{vmatrix}$
Apply $R_1\rightarrow R_1+R_2+R_3$
$\Delta=abc\begin{vmatrix}1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} & 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} &1+ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\\\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b}\\\frac{1}{c} & \frac{1}{c} &\frac{1}{c}+1\end{vmatrix}$
Take $(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$ as common factor from $R_1$
$\Delta=(abc)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\begin{vmatrix}1 & 1&1\\\frac{1}{b} & \frac{1}{b}+1 & \frac{1}{b}\\\frac{1}{c} & \frac{1}{c} &\frac{1}{c}+1\end{vmatrix}$
Now apply $C_2\rightarrow C_2-C_1,C_3\rightarrow C_3-C_1$
$\Delta=(abc)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\begin{vmatrix}1 & 0&0\\\frac{1}{b} & 1 & 0\\\frac{1}{c} & 0 &1\end{vmatrix}$
Now expanding along $R_1$ we get,
$\Delta=(abc)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})[1(1-0)]$
Therefore $\Delta=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$
$\qquad\qquad\;\;=abc+bc+ca+ab.$
Hence proved.
answered Mar 28, 2013 by sreemathi.v
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...