https://byjus.com/jee/differentiation-integration-of-determinants/
I saw this and I can't understand how this formula was derived, like why can we integrate row wise and add up determinants? Is there an intuition/ proof behind this?
https://byjus.com/jee/differentiation-integration-of-determinants/
I saw this and I can't understand how this formula was derived, like why can we integrate row wise and add up determinants? Is there an intuition/ proof behind this?
Hint: expanding $$ \det \left( \begin{matrix} f_1(x) & g_1(x) \\ f_2(x) & g_2(x) \end{matrix} \right) = f_1(x)g_2(x)-f_2(x)g_1(x) $$ and differentiating, $$f_1'(x)g_2(x)+f_1(x)g_2'(x)-f_2'(x)g_1(x)-f_2(x)g_1'(x),$$ you get the derivative of $\det \left( \begin{matrix} f_1(x) & g_1(x) \\ f_2(x) & g_2(x) \end{matrix} \right)$. You can rewrite the result as $$ \det \left( \begin{matrix} f_1'(x) & g_1'(x) \\ f_2(x) & g_2(x) \end{matrix} \right)+ \det \left( \begin{matrix} f_1(x) & g_1(x) \\ f_2'(x) & g_2'(x) \end{matrix} \right). $$ Use the same idea with integration.
By the Laplace expansion formula, a determinant can be expressed as a linear combination of the elements of a row (where the coefficients are the cofactors).
Hence by linearity of integration, you can integrate a row when the other rows are constants.