Using properties of determinants, prove that: 1 + a & 1 & 1 0.3em] 1 & 1 + b & 1 0.3em]1 & 1 &1 + c = abc + bc + ca + ab

R_1_ 1 a R_ 1, R_2 1 b R_ 2, R_3 1 c R_3 LHS =abc 1 a +1 & 1 a & 1 a 0.3em] 1 b & 1 b +1&1 b 0.3em] 1 c & 1 c &1 c +1 R_1 R_1 + R_ 2 + R_3 LHS =abc 1+1 a +1 b +1 c &1+1 a +1 b +1 c & 1+1 a +1 b +1 c 0.3em] 1 b & 1 b +1&1 b 0.3em] 1 c & 1 c &1 c +1 =abc ( 1+1 a +1 b +1 c ) 1 & 1& 1 0.3em] 1 b & 1 b +1&1 b 0.3em] 1 c & 1 c &1 c +1 c_2 _2-c_1 c_3 _3-c_1 =abc ( 1+1 a +1 b +1 c ) 1&0&0 1 b &1&0 1 c &0&1 =abc ( 1+1 a +1 b +1 c ).1 = abc + bc + ca + ab = RHS

Using properties of determinants, prove that: 1 + a & 1 & 1 0.3em…

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Question

Using properties of determinants, prove that:
$\begin{vmatrix} \text{1 + a } & \text{1} & \text{1} 0.3em] \text{1} & \text{1 + b} & \text{1} 0.3em]\text{1} & 1 &\text{1 + c} \end{vmatrix}= \text{ abc + bc + ca + ab}$

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Answer

$R_1$_$\rightarrow$
$\frac{1}{\text{a}}R_{1,}R_2$
$\rightarrow \frac{1}{\text{b}}R_{2,}R_3$
$\rightarrow \frac{1}{\text{c}}R_3$
$\therefore\ \text{LHS} =\text{abc} \begin{bmatrix} \frac{1}{\text{a}}+1 & \frac{1}{\text{a}}& \frac{1}{\text{a}} 0.3em] \frac{1}{\text{b}} & \frac{1}{\text{b}}+1&\frac{1}{\text{b}} 0.3em] \frac{1}{\text{c}} & \frac{1}{\text{c}} &\frac{1}{\text{c}}+1 \end{bmatrix}
R_1$
$\rightarrow R_1 + R_{2 +}R_3$
$\Rightarrow$ $\text{ LHS} =\text{abc} \begin{vmatrix} 1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}} &1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}& 1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}} 0.3em] \frac{1}{\text{b}} & \frac{1}{\text{b}}+1&\frac{1}{\text{b}} 0.3em] \frac{1}{\text{c}} & \frac{1}{\text{c}} &\frac{1}{\text{c}}+1 \end{vmatrix}$
$=\text{abc}\Bigg( 1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}\Bigg)$$\begin{vmatrix} 1 & 1& 1 0.3em] \frac{1}{\text{b}} & \frac{1}{\text{b}}+1&\frac{1}{\text{b}} 0.3em] \frac{1}{\text{c}} & \frac{1}{\text{c}} &\frac{1}{\text{c}}+1 \end{vmatrix}$
$\begin{matrix} \text{c}_2\rightarrow\text{c}_2-\text{c}_1$
$\text{c}_3\rightarrow\text{c}_3-\text{c}_1 \end{matrix}=\text{abc}\Bigg( 1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}\Bigg)\begin{vmatrix} 1&0&0$
$\frac{1}{\text{b}}&1&0$
$\frac{1}{\text{c}}&0&1$
$\end{vmatrix}$
$=\text{abc}\Bigg( 1+\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}\Bigg)$.1 = $\text{ abc + bc + ca + ab}$ = RHS

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