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

$\mathrm{M}_{11}=-12, \mathrm{M}_{21}=-16, \mathrm{M}_{31}=-4, \\ \mathrm{C}_{11}=-12, \mathrm{C}_{21}=16, \mathrm{C}_{31}=-4,$

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:

$\begin{gathered} \mathrm{A}=\left[\begin{array}{ccc} 1 & -3 & 2 \\ 4 & -1 & 2 \\ 3 & 5 & 2 \end{array}\right] \\ \end{gathered}$

Solution:

$\begin{gathered} \mathrm{A}=\left[\begin{array}{ccc} 1 & -3 & 2 \\ 4 & -1 & 2 \\ 3 & 5 & 2 \end{array}\right] \\ \mathrm{M}_{11}=\left|\begin{array}{cc} -1 & 2 \\ 5 & 2 \end{array}\right|=-2-10=-12 \\ \mathrm{M}_{21}=\left|\begin{array}{cc} -3 & 2 \\ 5 & 2 \end{array}\right|=-6-10=-16 \\ \mathrm{M}_{31}=\left|\begin{array}{ll} -3 & 2 \\ -1 & 2 \end{array}\right|=-6-(-2)=-4 \end{gathered}$

\begin{aligned} &\mathrm{C}_{\mathrm{ij}}=(-1)^{\mathrm{i}+\mathrm{j}} \mathrm{M}_{\mathrm{ij}} \\ &\mathrm{C}_{11}=(-1)^{2}(-12)=-12 \\ &\mathrm{C}_{21}=(-1)^{3}(-16)=16 \\ &\mathrm{C}_{31}=(-1)^{4}(-4)=-4 \\ &\mathrm{D}=1(-12)+3(8-6)+2(20+3) \\ &=-12+3(2)+2(23) \\ &\qquad \begin{array}{l} = -12+6+46 \\ =40 \end{array} \\ & \end{aligned}