Chapter 3 - Determinants - 3.2 Properties of Determinants - Problems - Page 221: 49

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According to property P6 we have $\begin{bmatrix} a_1+\beta b_1 &b_1+\gamma c_1 &c_1+\alpha a_1\\ a_2+\beta b_2 &b_2+\gamma c_2 &c_2+\alpha a_2\\ a_3+\beta b_3 &b_3+\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}\\ =\begin{bmatrix} a_1 &b_1+\gamma c_1 &c_1+\alpha a_1\\ a_2&b_2+\gamma c_2 &c_2+\alpha a_2\\ a_3&b_3+\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &b_1+\gamma c_1 &c_1+\alpha a_1\\ \beta b_2 &b_2+\gamma c_2 &c_2+\alpha a_2\\ \beta b_3 &b_3+\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}\\ =\begin{bmatrix} a_1 &b_1 &c_1+\alpha a_1\\ a_2&b_2&c_2+\alpha a_2\\ a_3&b_3 &c_3+\alpha a_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &\gamma c_1 &c_1+\alpha a_1\\ \beta b_2 &\gamma c_2 &c_2+\alpha a_2\\ \beta b_3 &\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}+\begin{bmatrix} a_1 &\gamma c_1 &c_1+\alpha a_1\\ a_2&\gamma c_2 &c_2+\alpha a_2\\ a_3&\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &\gamma c_1 &c_1+\alpha a_1\\ \beta b_2 &\gamma c_2 &c_2+\alpha a_2\\ \beta b_3 &\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}\\ =\begin{bmatrix} a_1 &b_1 &c_1\\ a_2&b_2&c_2\\ a_3&b_3 &c_3 \end{bmatrix}+\begin{bmatrix} a_1 &\gamma c_1 &c_1\\ a_2 &\gamma c_2 &c_2\\ a_3 &\gamma c_3 &c_3 \end{bmatrix}+\begin{bmatrix} a_1 &b_1 & \alpha a_1\\ a_2& b_2 & \alpha a_2\\ a_3 & b_3 &\alpha a_3 \end{bmatrix}+\begin{bmatrix} a_1 &\gamma c_1 &\alpha a_1\\ a_2 &\gamma c_2 & \alpha a_2\\ a_3 &\gamma c_3 & \alpha a_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &b_1 &c_1\\ \beta b_2&b_2&c_2\\ \beta b_3&b_3 &c_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &\gamma c_1 &c_1\\ \beta b_2 &\gamma c_2 &c_2\\ \beta b_3 &\gamma c_3 &c_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &b_1 & \alpha a_1\\ \beta b_2& b_2 & \alpha a_2\\ \beta b_3 & b_3 &\alpha a_3 \end{bmatrix}+\begin{bmatrix} \beta b_1 &\gamma c_1 &\alpha a_1\\ \beta b_2 &\gamma c_2 & \alpha a_2\\ \beta b_3 &\gamma c_3 & \alpha a_3 \end{bmatrix}$ Using property 4 $=\begin{bmatrix} a_1 &b_1 &c_1\\ a_2&b_2&c_2\\ a_3&b_3 &c_3 \end{bmatrix}+\gamma \begin{bmatrix} a_1 & c_1 &c_1\\ a_2 & c_2 &c_2\\ a_3 & c_3 &c_3 \end{bmatrix}+\alpha \begin{bmatrix} a_1 &b_1 & a_1\\ a_2& b_2 & a_2\\ a_3 & b_3 &a_3 \end{bmatrix}+\gamma \alpha\begin{bmatrix} a_1 & c_1 &a_1\\ a_2 &c_2 & a_2\\ a_3 &c_3 & a_3 \end{bmatrix}+\beta \begin{bmatrix} b_1 &b_1 &c_1\\ b_2&b_2&c_2\\ b_3&b_3 &c_3 \end{bmatrix}+\beta \gamma \begin{bmatrix} b_1 & c_1 &c_1\\ b_2 & c_2 &c_2\\ b_3 & c_3 &c_3 \end{bmatrix}+\beta \alpha \begin{bmatrix} b_1 &b_1 & a_1\\ b_2& b_2 & a_2\\ b_3 & b_3 & a_3 \end{bmatrix}+\beta \gamma \alpha \begin{bmatrix} b_1 & c_1 & a_1\\ b_2 & c_2 & a_2\\ b_3 & c_3 & a_3 \end{bmatrix}$ Hence, the determinant is: $\begin{bmatrix} a_1+\beta b_1 &b_1+\gamma c_1 &c_1+\alpha a_1\\ a_2+\beta b_2 &b_2+\gamma c_2 &c_2+\alpha a_2\\ a_3+\beta b_3 &b_3+\gamma c_3 &c_3+\alpha a_3 \end{bmatrix}\\= \begin{vmatrix} a_1 &b_1 & a_1\\ a_2& b_2 & a_2\\ a_3 & b_3 &a_3 \end{vmatrix}+\alpha \beta \gamma \begin{vmatrix} b_1 & c_1 & a_1\\ b_2 & c_2 & a_2\\ b_3 & c_3 & a_3 \end{vmatrix}\\ =(1+\alpha \beta \gamma)\begin{vmatrix} b_1 & c_1 & a_1\\ b_2 & c_2 & a_2\\ b_3 & c_3 & a_3 \end{vmatrix}$ The given determinant is zero for all $a_i,b_i,c_i$ if and only if: $1+\alpha \beta \gamma =0\\ \alpha \beta \gamma =-1$