Differentiation and Integration of Determinants

Differentiating and integrating determinants is one of the integral concepts in mathematics. This lesson will cover the steps on how to differentiate and integrate determinants easily using several solved example questions.

All Contents in Determinants

Differentiation of Determinants

Let Δ(x)=f1(x)g1(x)f2(x)g2(x),    where    f1(x),f2(x),g1(x)    and    g2(x)\Delta \left( x \right)=\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|,\;\;where \;\;{{f}_{1}}\left( x \right),{{f}_{2}}\left( x \right),{{g}_{1}}\left( x \right)\;\; and \;\;{{g}_{2}}\left( x \right)

are functions of x. Then,

Δ(x)=f1(x)g1(x)f2(x)g2(x)+f1(x)g1(x)f2(x)g2(x)      Also,      Δ(x)=f1(x)g1(x)f2(x)g2(x)+f1(x)g1(x)f2(x)g2(x)\Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{g}_{1}}’\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|+\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}’\left( x \right) & {{g}_{2}}’\left( x \right) \\ \end{matrix} \right| \;\;\;Also, \;\;\;\Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{g}_{1}}\left( x \right) \\ {{f}_{2}}’\left( x \right) & {{g}_{2}}\left( x \right) \\ \end{matrix} \right|+\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{g}_{1}}’\left( x \right) \\ {{f}_{2}}\left( x \right) & {{g}_{2}}’\left( x \right) \\ \end{matrix} \right|

How to Differentiate a Determinant?

Thus, to differentiate a determinant, we differentiate one row (or column) at a time, keeping others unchanged. If we write Δ(x)=[C1C2],\Delta \left( x \right)=\left[ {{C}_{1}}\,\,\,{{C}_{2}} \right], where Ci denotes the ith column, then Δ(x)=[C1C2]+[C1C2],\Delta ‘\left( x \right)=\left[ {{C}_{1}}’\,\,\,{{C}_{2}} \right]+\left[ {{C}_{1}}\,\,\,{{C}_{2}}’ \right], where Ci denotes the column obtained by differentiating functions in the ith column Ci Also, if Δ(x)=[R1R2],  then  Δ(x)=[R1R2]+[R1R2]\Delta \left( x \right)=\left[ \begin{matrix} {{R}_{1}} \\ {{R}_{2}} \\ \end{matrix} \right],\; then \;\Delta ‘\left( x \right)=\left[ \begin{matrix} {{R}_{1}}’ \\ {{R}_{2}} \\ \end{matrix} \right]+\left[ \begin{matrix} {{R}_{1}} \\ {{R}_{2}}’ \\ \end{matrix} \right]

Similarly, we can differentiate determinants of higher order.

Note: Differentiation can also be done column-wise by taking one column at a time.

Integration of Deteminants

If f(x), g(x) and h(x) are functions of x and a, b, c, α, β and γ are constants such that

Δ(x)=f(x)g(x)h(x)abcαβγ,\Delta \left( x \right)=\left| \begin{matrix} f\left( x \right) & g\left( x \right) & h\left( x \right) \\ a & b & c \\ \alpha & \beta & \gamma \\ \end{matrix} \right|,

then the integral of the determinants i.e. Δ(x)  is  given  by  Δ(x)dx=f(x)dxg(x)dxh(x)dxabcαβγ\Delta \left( x \right)\; is\; given\; by \; \int{\Delta \left( x \right)dx=\left| \begin{matrix} \int{f\left( x \right)dx} & \int{g\left( x \right)dx} & \int{h\left( x \right)dx} \\ a & b & c \\ \alpha & \beta & \gamma \\ \end{matrix} \right|}

Example Problems on Differentiation and Integration of Determinants

Example 1: If Δ(x)=sin2xlogcosxlogtanxn22n12n+112log20,  then  evaluate  0π/2Δ(x)dx.\Delta \left( x \right)=\left| \begin{matrix} {{\sin }^{2}}x & \log \cos x & \log \tan x \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|,\; then \;evaluate\; \int\limits_{0}^{\pi /2}{\Delta \left( x \right)dx.}

Solution:

By applying integration on variable elements of determinant we will solve the given problem.

We have Δ(x)=sin2xlogcosxlogtanxn22n12n+112log20;0π/2Δ(x)dx=0π/2sin2xdx0π/2logcosxdx0π/2logtanxdxn22n12n+112log20\Delta \left( x \right)=\left| \begin{matrix} {{\sin }^{2}}x & \log \cos x & \log \tan x \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|;\,\,\int\limits_{0}^{\pi /2}{\Delta \left( x \right)dx=\left| \begin{matrix} \int\limits_{0}^{\pi /2}{{{\sin }^{2}}x\,dx} & \int\limits_{0}^{\pi /2}{\log \,\cos x\,dx} & \int\limits_{0}^{\pi /2}{\log \tan x\,dx} \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|} =π4π2log20n22n12n+112log20=π412log20n22n12n+112log20=π4×0=0=\left| \begin{matrix} \frac{\pi }{4} & -\frac{\pi }{2}\log 2 & 0 \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|=\frac{\pi }{4}\left| \begin{matrix} 1 & -2\log 2 & 0 \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\ \end{matrix} \right|=\frac{\pi }{4}\times 0=0

 

Example 2: If f(x)=xnsinxcosxn!sinnπ2cosnπ2aa2a3,  then  show  that  dndxn{f(x)}=0  at    x=0.f\left( x \right)=\left| \begin{matrix} {{x}^{n}} & \sin x & \cos x \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|,\; then \;show \;that \;\frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\}=0 \;at \;\;x=0.

Solution:

By applying integration on variable elements of the determinant we will solve the given problem.

We have, f(x)=xnsinxcosxn!sinnπ2cosnπ2aa2a3;    dndxn{f(x)}=dndxn(xn)dndxn(sinx)dndxn(cosx)n!sinnπ2cosnπ2aa2a3f\left( x \right)=\left| \begin{matrix} {{x}^{n}} & \sin x & \cos x \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|; \;\;\frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\}=\left| \begin{matrix} \frac{{{d}^{n}}}{d{{x}^{n}}}\left( {{x}^{n}} \right) & \frac{{{d}^{n}}}{d{{x}^{n}}}\left( \sin x \right) & \frac{{{d}^{n}}}{d{{x}^{n}}}\left( \cos x \right) \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right| =n!sin(x+nπ2)cos(x+nπ2)n!sinnπ2cosnπ2aa2a3;        (dndxn{f(x)})x=0=n!sinnπ2cos+nπ2n!sinnπ2cosnπ2aa2a3=0=\left| \begin{matrix} n! & \sin \left( x+\frac{n\pi }{2} \right) & \cos \left( x+\frac{n\pi }{2} \right) \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|; \;\;\;\;{{\left( \frac{{{d}^{n}}}{d{{x}^{n}}}\left\{ f\left( x \right) \right\} \right)}_{x=0}}=\left| \begin{matrix} n! & \sin \frac{n\pi }{2} & \cos +\frac{n\pi }{2} \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix} \right|=0

Problem Solving Tactics

Let Δ(x)=f1(x)f2(x)f3(x)b1b2b3c1c2c3,      then      Δ(x)=f1(x)f2(x)f3(x)b1b2b3c1c2c3,\Delta \left( x \right)=\left| \begin{matrix} {{f}_{1}}\left( x \right) & {{f}_{2}}\left( x \right) & {{f}_{3}}\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|,\;\;\; then \;\;\; \Delta ‘\left( x \right)=\left| \begin{matrix} {{f}_{1}}’\left( x \right) & {{f}_{2}}’\left( x \right) & {{f}_{3}}’\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|,

and in general

Δn(x)=f1n(x)f2n(x)f3n(x)b1b2b3c1c2c3{{\Delta }^{n}}\left( x \right)=\left| \begin{matrix} f_{1}^{n}\left( x \right) & f_{2}^{n}\left( x \right) & f_{3}^{n}\left( x \right) \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right| where n is any positive integer and fn(x){{f}^{n}}\left( x \right) denotes the nth derivative of f(x).f\left( x \right).

Let Δ(x)=f(x)g(x)h(x)abclmn,\Delta \left( x \right)=\left| \begin{matrix} f\left( x \right) & g\left( x \right) & h\left( x \right) \\ a & b & c \\ l & m & n \\ \end{matrix} \right|,

where a,b,c,l,m and n are constants.

abΔ(x)dx=abf(x)dxabg(x)dxabh(x)dxabclmn\Rightarrow \,\,\int\limits_{a}^{b}{\Delta \left( x \right)dx=\left| \begin{matrix} \int\limits_{a}^{b}{f\left( x \right)dx} & \int\limits_{a}^{b}{g\left( x \right)dx} & \int\limits_{a}^{b}{h\left( x \right)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 onlyafter evaluation/expansion of the determinant.