In a determinant prove that $a_2A_1 + b_2B_1 + c_2C_1 = 0$ where $A_1, B_1, C_1$ is the cofactors of $a_1, b_1, c_1.$?

Question

Here is the determinant: $$ \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\\ \end{vmatrix} $$

in the determinat prove that $a_2A_1 + b_2B_1 + c_2C_1 = 0$ where $A_1, B_1, C_1$ is the cofactors of $a_1, b_1, c_1.$

P.S: I am a total beginner in matrices and determinants.This is a whole new weird question to me.Please explain the answers.

Answer 1

According to the definition of cofactors of the elements of the original matrix you would have $$a_1A_1+b_1B_1+c_1C_1={\rm det}(A)\ .$$ Now you have replaced $(a_1,b_1,c_1)$ in this formula by $(a_2,b_2,c_2)$. This means that you compute the determinant of the matrix $$\left[\matrix{a_2&b_2&b_2\cr a_2&b_2&c_2\cr a_3&b_3&c_3\cr}\right]$$ having two equal lines.

Answer 2

You will always get $0$, since it's equivalent to computing the determinant of a matrix with duplicate rows.