Please solve RD Sharma class 12 chapter Determinants exercise 5.1 question 1 subquestion (v) maths textbook solution

$\mathrm{M}_{11}=5, \mathrm{M}_{21}=-40, \mathrm{M}_{31}=-30 \\ \mathrm{C}_{11}=5, \mathrm{C}_{21}=40, \mathrm{C}_{31}=-30$

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} &\mathbf{A}=\left[\begin{array}{lll} 0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1 \end{array}\right] \\ \end{aligned}

Solution:

\begin{aligned} &\mathbf{A}=\left[\begin{array}{lll} 0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1 \end{array}\right] \\ &\mathrm{M}_{11}=\left|\begin{array}{ll} 5 & 0 \\ 7 & 1 \end{array}\right|=5-0=5 \\ &\mathrm{M}_{21}=\left|\begin{array}{ll} 2 & 6 \\ 7 & 1 \end{array}\right|=2-42=-40 \\ &\mathrm{M}_{31}=\left|\begin{array}{ll} 2 & 6 \\ 5 & 0 \end{array}\right|=0-(30)=-30 \end{aligned}

\begin{aligned} &C_{i j}=(-1)^{\mathrm{i}+j} \mathrm{M}_{\mathrm{ij}} \\ &\mathrm{C}_{11}=(-1)^{2} 5=5 \\ &\mathrm{C}_{21}=(-1)^{3}(-40)=40 \\ &\mathrm{C}_{31}=(-1)^{4}(-30)=-30 \\ &\mathrm{D}=0(5)-2(1)+6(7-5) \\ &=0-2-48 \\ &=-50 \end{aligned}