- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.6. $\quad$ Exercises
7. $\quad$ State and prove a generalization of Exercise 6 for the determinant $$F(x) = \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ g_1(x) & g_2(x) & g_3(x) \\ h_1(x) & h_2(x) & h_3(x) \end{vmatrix}.$$
Solution. $\quad$ We know that determinants of order three may be computed by Equation (3.2). \begin{align*} (3.2) \qquad \det\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} &= a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}. \end{align*} Accordingly, this gives us \begin{align*} F(x) &= f_1(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2(x) & h_3(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1(x) & h_3(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1(x) & g_1(x) \\ h_1(x) & h_2(x) \end{vmatrix}. \end{align*} Using the product rule for derivatives, we have \begin{align*} F'(x) &= f_1'(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2(x) & h_3(x) \end{vmatrix} + f_1(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2(x) & h_3(x) \end{vmatrix}' \\ &- f_2'(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1(x) & h_3(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1(x) & h_3(x) \end{vmatrix}' \\ &+ f_3'(x)\begin{vmatrix} g_1(x) & g_1(x) \\ h_1(x) & h_2(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1(x) & g_1(x) \\ h_1(x) & h_2(x) \end{vmatrix}' \end{align*} Then, applying the result of Exercise 6, we get the following generalization for the determinant $F(x):$ \begin{align*} F'(x) &= f_1'(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2(x) & h_3(x) \end{vmatrix} + f_1(x)\begin{vmatrix} g_2'(x) & g_3'(x) \\ h_2(x) & h_3(x) \end{vmatrix} + f_1(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2'(x) & h_3'(x) \end{vmatrix} \\ &- f_2'(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1(x) & h_3(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1'(x) & g_3'(x) \\ h_1(x) & h_3(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1'(x) & h_3'(x) \end{vmatrix} \\ &+ f_3'(x)\begin{vmatrix} g_1(x) & g_2(x) \\ h_1(x) & h_2(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1'(x) & g_2'(x) \\ h_1(x) & h_2(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1(x) & g_2(x) \\ h_1'(x) & h_2'(x) \end{vmatrix} \\ \\ &= f_1'(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2(x) & h_3(x) \end{vmatrix} - f_2'(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1(x) & h_3(x) \end{vmatrix} + f_3'(x)\begin{vmatrix} g_1(x) & g_2(x) \\ h_1(x) & h_2(x) \end{vmatrix} \\ &+ f_1(x)\begin{vmatrix} g_2'(x) & g_3'(x) \\ h_2(x) & h_3(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1'(x) & g_3'(x) \\ h_1(x) & h_3(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1'(x) & g_2(x) \\ h_1(x) & h_2(x) \end{vmatrix} \\ &+ f_1(x)\begin{vmatrix} g_2(x) & g_3(x) \\ h_2'(x) & h_3'(x) \end{vmatrix} - f_2(x)\begin{vmatrix} g_1(x) & g_3(x) \\ h_1'(x) & h_3'(x) \end{vmatrix} + f_3(x)\begin{vmatrix} g_1(x) & g_2(x) \\ h_1(x) & h_2(x) \end{vmatrix} \\ \\ &= \begin{vmatrix} f_1'(x) & f_2'(x) & f_3'('x) \\ g_1(x) & g_2(x) & g_3(x) \\ h_1(x) & h_2(x) & h_3(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ g_1'(x) & g_2'(x) & g_3'(x) \\ h_1(x) & h_2(x) & h_3(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & f_2(x) & f_3(x) \\ g_1(x) & g_2(x) & g_3(x) \\ h_1'(x) & h_2'(x) & h_3'(x) \end{vmatrix}. \quad \blacksquare \end{align*}