JEE Main & Advanced Mathematics Determinants & Matrices Product of Two Determinants

 Let the two determinants of third order be,

\[{{D}_{1}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\] and \[{{D}_{2}}=\left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\].

Let D be their product.

Then \[D=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\,\,\,\times \left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\]

\[=\left| \,\begin{matrix} {{a}_{1}}{{\alpha }_{1}}+{{b}_{1}}{{\beta }_{1}}+{{c}_{1}}{{\gamma }_{1}} & {{a}_{1}}{{\alpha }_{2}}+{{b}_{1}}{{\beta }_{2}}+{{c}_{1}}{{\gamma }_{2}} & {{a}_{1}}{{\alpha }_{3}}+{{b}_{1}}{{\beta }_{3}}+{{c}_{1}}{{\gamma }_{3}}  \\ {{a}_{2}}{{\alpha }_{1}}+{{b}_{2}}{{\beta }_{1}}+{{c}_{2}}{{\gamma }_{1}} & {{a}_{2}}{{\alpha }_{2}}+{{b}_{2}}{{\beta }_{2}}+{{c}_{2}}{{\gamma }_{2}} & {{a}_{2}}{{\alpha }_{3}}+{{b}_{2}}{{\beta }_{3}}+{{c}_{2}}{{\gamma }_{3}}  \\ {{a}_{3}}{{\alpha }_{1}}+{{b}_{3}}{{\beta }_{1}}+{{c}_{3}}{{\gamma }_{1}} & {{a}_{3}}{{\alpha }_{2}}+{{b}_{3}}{{\beta }_{2}}+{{c}_{3}}{{\gamma }_{2}} & {{a}_{3}}{{\alpha }_{3}}+{{b}_{3}}{{\beta }_{3}}+{{c}_{3}}{{\gamma }_{3}}  \\ \end{matrix}\, \right|\]

We can also multiply rows by columns or columns by rows or columns by columns.