$\Delta_{1}=\left|\begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right| \text { and } \Delta_{2}=\left|\begin{array}{ccc}{\alpha_{1}} & {\beta_{1}} & {\gamma_{1}} \\ {\alpha_{2}} & {\beta_{2}} & {\gamma_{2}} \\ {\alpha_{3}} & {\beta_{3}} & {\gamma_{3}}\end{array}\right|$

We can multiply these row-by-row or column-by-column or row-by-column or column-by-row

Row-by-row multiplication of these two determinants is given by

$\mathrm{\Delta_1\times \Delta_2=\begin{vmatrix} a_1\alpha _1+b_1\beta _1+c_1\gamma _1& a_1\alpha _2+b_1\beta _2+c_1\gamma _2 &a_1\alpha _3+b_1\beta _3+c_1\gamma _3 \\ a_2\alpha _1+b_2\beta _1+c_2\gamma _1 & a_2\alpha _2+b_2\beta _2+c_2\gamma _2 &a_2\alpha _3+b_2\beta _3+c_2\gamma _3 \\ a_3\alpha _1+b_3\beta _1+c_3\gamma _1& a_3\alpha _2+b_3\beta _2+c_3\gamma _2 & a_3\alpha _3+b_3\beta _3+c_3\gamma _3 \end{vmatrix}}$

Multiplication can also be performed row by column; column by row or column by column as required in the problem.

To express a determinant as a product of two determinants, one requires lots of practice and this can be done only by inspection and trial.

Property:

$\\\mathrm{If\:A_1,\:B_1,\:C_1,\:.....are\:respectively\:the\:cofactors\:of\:the\:elements\:a_1,\:b_1,\:c_1\:..........\:of\:the\:determinant}\\\mathrm{\Delta=\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2& b_2 & c_2\\ a_3 &b_3 & c_3 \end{vmatrix},\;\Delta\neq0,\;\;then\;\;\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2& B_2 & C_2\\ A_3 &B_3 & C_3 \end{vmatrix}}=\Delta^2$

Proof :

$\\\mathrm{given,\;\;\Delta=\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2& b_2 & c_2\\ a_3 &b_3 & c_3 \end{vmatrix} }\\\mathrm{and,\;A_1,\:B_1,\:C_1,\:.....\;are\:respectively\:the\:cofactors\:of\:the\:elements\:a_1,\:b_1,\:c_1\:..........\:Hence,}\\\\\mathrm{\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2& b_2 & c_2\\ a_3 &b_3 & c_3 \end{vmatrix}\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2& B_2 & C_2\\ A_3 &B_3 & C_3 \end{vmatrix} }\\\\\mathrm{=\begin{vmatrix}\:a_1A\:_1+b_1B\:_1+c_1C\:_1&\:a_1A\:_2+b_1B\:_2+c_1C\:_2\:&a_1A\:_3+b_1B\:_3+c_1C\:_3\:\\ \:a_2A\:_1+b_2B\:_1+c_2C\:_1\:&\:a_2A\:_2+b_2B\:_2+c_2C\:_2\:&a_2A\:_3+b_2B\:_3+c_2C\:_3\:\\ \:\:a_3A\:_1+b_3B\:_1+c_3C\:_1&\:a_3A\:_2+b_3B\:_2+c_3C\:_2\:&\:a_3A\:_3+b_3B\:_3+c_3C\:_3\end{vmatrix}}\\\\\mathrm{[row\;by\;row\;multiplication]}\\\mathrm{=\begin{vmatrix}\:\Delta &0&0\\ \:0\:&\:\Delta \:&0\\ 0&\:0\:&\:\Delta \:\end{vmatrix}=\Delta^3}\\\mathrm{\because \;a_iA_j+b_iB_j+c_iC_j=\left\{\begin{matrix} \Delta, &i=j \\ 0,&i\neq j \end{matrix}\right.}$
$\\\mathrm{\Rightarrow \Delta\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2& B_2 & C_2\\ A_3 &B_3 & C_3 \end{vmatrix} =\Delta^3}\\\\\mathrm{\Rightarrow \begin{vmatrix} A_1 &B_1 &C_1 \\ A_2& B_2 & C_2\\ A_3 &B_3 & C_3 \end{vmatrix} }=\Delta^2$

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