#### Provide solution for RD Sharma maths class 12 chapter Determinants exercise 5.1 question 1 subquestion (vi)

\begin{aligned} &\mathrm{M}_{11}=\mathrm{bc}-\mathrm{f}^{2}, \mathrm{M}_{21}=\mathrm{hc}-\mathrm{fg}, \mathrm{M}_{31}=\mathrm{hf}-\mathrm{gb} \\ &\mathrm{C}_{11}=\mathrm{bc}-\mathrm{f}^{2}, \mathrm{C}_{21}=-(\mathrm{hc}-\mathrm{fg}), \mathrm{C}_{31}=\mathrm{hf}-\mathrm{gb} \end{aligned}

Hint:

Let Mij, Cij represents minor and cofactor of an element, where i and j represent the row and column. The minor of matrix can be obtained for particular element by removing the row and column where the element is present.

Given:

\begin{aligned} &\mathrm{A}=\left[\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right] \\ \end{aligned}

Solution:

\begin{aligned} &\mathrm{A}=\left[\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right] \\ &\mathrm{M}_{11}=\left|\begin{array}{ll} b & f \\ f & c \end{array}\right|=\mathrm{bc}-\mathrm{f}^{2} \\ &\mathrm{M}_{21}=\left|\begin{array}{ll} h & g \\ f & c \end{array}\right|=\mathrm{hc}-\mathrm{fg} \\ &\mathrm{M}_{31}=\left|\begin{array}{ll} h & g \\ b & f \end{array}\right|=\mathrm{hf}-\mathrm{gb} \end{aligned}

\begin{aligned} &C_{i j}=(-1)^{i+j} M_{i j} \\ &C_{11}=(-1)^{2} b c-f^{2}=b c-f^{2} \\ &C_{21}=(-1)^{3} \mathrm{hc}-f g=-(h c-f g) \\ &C_{31}=(-1)^{4} \mathrm{hf}-g b=h f-g b \end{aligned}

\begin{aligned} \mathrm{D} &=\mathrm{a}\left(\mathrm{b} \mathrm{c}-\mathrm{f}^{2}\right)-\mathrm{h}(\mathrm{hc}-\mathrm{fg})+\mathrm{g}(\mathrm{hf}-\mathrm{g} \mathrm{b}) \\ &=\mathrm{abc}-\mathrm{af}^{2}-\mathrm{h}^{2} \mathrm{c}+\mathrm{hfg}+\mathrm{ghf}-\mathrm{g}^{2} \mathrm{~b} \\ &=\mathrm{abc}-\mathrm{af}^{2}-\mathrm{h}^{2} \mathrm{c}+2 \mathrm{hfg}-\mathrm{g}^{2} \mathrm{~b} \end{aligned}