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

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

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

= \(\frac{\pi }{4}\times 0=0\).