Question

Consider the determinant $\Delta =\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

$M_{ij}=$ Minor of the element of $i^{th}$ row & $j^{th}$ column.

$C_{ij}=$ Cofactor of element of $i^{th}$ row & $j^{th}$ column.

$a_3{M}_{13}-b_3{M}_{23}+c_3{M}_{33}$ is equal to

• A. $0$
• B. $4\Delta$
• C. $2\Delta$
• D. $\Delta$

Answer 1

The correct option is D $\Delta$

$a_3{M}_{13}-b_2{M}_{23}+c_3{M}_{33}$

$=a_3\left|\begin{array}{cc}{b}_1& b_2\\ c_1& c_2\end{array}\right|-b_3\left|\begin{array}{cc}{a}_1& a_2\\ c_1& c_2\end{array}\right|+c_3\left|\begin{array}{cc}{a}_1& a_2\\ b_1& b_2\end{array}\right|$

Is equal to the expansion of $△$ along $C_3$