# Limit, Differentiation and Integration of Determinant

Limit of a determinant: Let $$\Delta (x)=\left| \begin{matrix}f(x) & g(x) & h(x) \\l(x) &m(x) & n(x) \\u(x) & v(x) & w(x) \\\end{matrix} \right|\,$$ then

$$\underset{x\to a}{\mathop{\lim }}\,\Delta(x)=\left| \begin{matrix}\underset{x\to a}{\mathop{\lim }}\,f(x) & \underset{x\to a}{\mathop{\lim }}\,g(x) & \underset{x\to a}{\mathop{\lim }}\,h(x) \\\underset{x\to a}{\mathop{\lim }}\,l(x) & \underset{x\to a}{\mathop{\lim }}\,m(x) &\underset{x\to a}{\mathop{\lim }}\,n(x) \\\underset{x\to a}{\mathop{\lim }}\,u(x) &\underset{x\to a}{\mathop{\lim }}\,v(x) & \underset{x\to a}{\mathop{\lim }}\,w(x) \\\end{matrix} \right|$$ provided each of nine limiting values exist finitely.

Differentiation of a determinant: Let $$\Delta (x)=\left|\begin{matrix}f(x) & g(x) & h(x)\\l(x) & m(x) & n(x)\\u(x) & v(x) & w(x)\\\end{matrix}\right|$$ then

$$\Delta ‘(x)=\left|\begin{matrix}f'(x) & g'(x) & h'(x)\\l(x) & m(x) & n(x)\\u(x) & v(x) & w(x)\\\end{matrix}\right|+\left|\begin{matrix}f(x) & g(x) & h(x)\\l'(x) & m'(x) & n'(x)\\u(x) & v(x) & w(x)\\\end{matrix} \right|+\left|\begin{matrix}f(x) & g(x) & h(x)\\l(x) & m(x) & n(x)\\u'(x) & v'(x) & w'(x)\\\end{matrix}\right|\,$$.

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, b, c, l, m and n are constants then $$\underset{a}{\mathop{\overset{b}{\mathop{\int }}\,}}\,\Delta (x)=\left| \begin{matrix}\underset{a}{\mathop{\overset{b}{\mathop{\int }}\,}}\,f(x) &\underset{a}{\mathop{\overset{b}{\mathop{\int }}\,}}\,g(x) &\underset{a}{\mathop{\overset{b}{\mathop{\int }}\,}}\,h(x) \\a & b & c \\l & m & n \\\end{matrix} \right|$$.

If more than one row (column) of Δ (x) are variable, then in order to find $$\underset{a}{\mathop{\overset{b}{\mathop{\int }}\,}}\,\Delta (x)dx$$ first we evaluate the determinant Δ (x) by using the properties of determinants and we integrate it.

Ex: If $$\Delta (x)=\left| \begin{matrix}{{\sin }^{2}}x & \log \cos x & \log \tan x \\{{n}^{2}} & 2n-1 & 2n+1 \\1 & -2\log 2 & 0 \\\end{matrix}\right|$$, then evaluate $$\underset{a}{\mathop{\overset{\frac{\pi }{2}}{\mathop{\int }}\,}}\,\Delta (x)dx$$.

Sol: $$\underset{0}{\mathop{\overset{\frac{\pi }{2}}{\mathop{\int }}\,}}\,\Delta (x)dx=\left|\begin{matrix}\underset{0}{\mathop{\overset{\frac{\pi }{2}}{\mathop{\int}}\,}}\,{{\sin }^{2}}x & \underset{0}{\mathop{\overset{\frac{\pi}{2}}{\mathop{\int }}\,}}\,\log \cos x &\underset{0}{\mathop{\overset{\frac{\pi }{2}}{\mathop{\int }}\,}}\,\log \tan x \\ {{n}^{2}} & 2n-1 & 2n+1 \\ 1 & -2\log 2 & 0 \\\end{matrix} \right|$$.

= $$\left| \begin{matrix}\frac{\pi }{4} & -\frac{\pi }{2}\log 2 & 0 \\{{n}^{2}} & 2n-1 & 2n+1 \\1 & -2\log 2 & 0 \\\end{matrix} \right|$$.

= $$\frac{\pi }{4}\left| \begin{matrix}1 & -2\log 2 & 0 \\{{n}^{2}} & 2n-1 & 2n+1 \\1 & -2\log 2 & 0 \\\end{matrix} \right|$$.

= $$\frac{\pi }{4}\times 0=0$$.