JEE Main & Advanced Mathematics Determinants Differentiation and Integration of Determinants

Differentiation and Integration of Determinants

Category : JEE Main & Advanced

(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.

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