JEE Main & Advanced Mathematics Determinants & Matrices Differentiation and Integration of Determinants

(1) Differentiation of a determinant

(i) Let \[\Delta (x)\] be a determinant of order two. If we write \[\Delta (x)=|{{C}_{1}}\,\,\,\,\,{{C}_{2}}|\], where \[{{C}_{1}}\] and \[{{C}_{2}}\] denote the 1st and 2nd columns, then

\[\Delta '(x)=\left| \,\begin{matrix}  C{{'}_{1}} & {{C}_{2}}  \\ \end{matrix} \right|+\left| \,\begin{matrix} {{C}_{1}} & {{{{C}'}}_{2}}  \\ \end{matrix} \right|\]

where \[C{{'}_{i}}\] denotes the column which contains the derivative of all the functions in the \[{{i}^{th}}\]column \[{{C}_{i}}\].

In a similar fashion, if we write \[\Delta (x)=\left| \,\begin{matrix} {{R}_{1}}  \\ {{R}_{2}}  \\ \end{matrix}\, \right|\], then \[{\Delta }'\,(x)=\left| \,\begin{matrix} R{{'}_{1}}  \\ {{R}_{2}}  \\ \end{matrix}\, \right|\,+\,\left| \,\begin{matrix} {{R}_{1}}  \\ {{{{R}'}}_{2}}  \\ \end{matrix}\, \right|\,\]

(ii) Let \[\Delta (x)\] be a determinant of order three. If we write \[\Delta (x)=\left| \,\begin{matrix} {{C}_{1}} & {{C}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|\], then

\[\Delta '(x)=\left| \,\begin{matrix} C{{'}_{1}} & {{C}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|+\left| \,\begin{matrix} {{C}_{1}} & C{{'}_{2}} & {{C}_{3}}\,  \\ \end{matrix} \right|+\left| \,\begin{matrix} {{C}_{1}} & {{C}_{2}} & C{{'}_{3}}\,  \\ \end{matrix} \right|\]

and similarly if we consider \[\Delta (x)=\left| \,\begin{matrix} {{R}_{1}}  \\ {{R}_{2}}  \\ {{R}_{3}}  \\ \end{matrix}\, \right|\]

Then  \[\Delta '(x)=\left| \,\begin{matrix} R{{'}_{1}}  \\ {{R}_{2}}  \\ {{R}_{3}}  \\ \end{matrix}\, \right|+\left| \,\begin{matrix} {{R}_{1}}  \\ R{{'}_{2}}  \\ {{R}_{3}}  \\ \end{matrix}\, \right|+\left| \,\begin{matrix} {{R}_{1}}  \\ {{R}_{2}}  \\ R{{'}_{3}}  \\ \end{matrix}\, \right|\]

(iii) If only one row (or column) consists functions of \[x\] and other rows (or columns) are constant, viz.

Let \[\Delta (x)=\left| \,\begin{matrix} {{f}_{1}}(x) & {{f}_{2}}(x) & {{f}_{3}}(x)  \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix}\, \right|\], 

Then \[\Delta '(x)=\left| \,\begin{matrix} f{{'}_{1}}(x) & f{{'}_{2}}(x) & f{{'}_{3}}(x)  \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix}\, \right|\]

And in general \[{{\Delta }^{n}}(x)=\left| \,\begin{matrix} {{f}_{1}}^{n}(x) & {{f}_{2}}^{n}(x) & {{f}_{3}}^{n}(x)  \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix}\, \right|\]

where \[n\] is any positive integer and \[{{f}^{n}}(x)\] denotes the \[{{n}^{th}}\] derivative of \[f(x)\].

(2) Integration of a determinant

Let \[\Delta (x)=\left| \,\begin{matrix} f(x) & g(x) & h(x)  \\ a & b & c  \\ l & m & n  \\ \end{matrix}\, \right|\], where \[a,\text{ }b,\text{ }c,\text{ }l,\text{ }m\] and \[n\] are constants.

\[\Rightarrow \,\int_{a}^{b}{\Delta (x)dx=\left| \,\begin{matrix} \int_{a}^{b}{f(x)dx} & \int_{a}^{b}{g(x)dx} & \int_{a}^{b}{h(x)dx}  \\ a & b & c  \\ l & m & n  \\ \end{matrix}\, \right|}\]   

• If the elements of more than one column or rows are functions of \[x\] then the integration can be done only after evaluation/expansion of the determinant.