To find the determinant in this question

Question

Given $A$ by $4×4$ non singular matrix and $B$ be matrix obtained from A by adding to its third row twice the first row .Then $det(2A^{-1}B)$ is

$A:2$

$B:4$

$C:8$

$D:16$

I cannot think anything about this question .Any hints to get started

Answer 1

let $A=|R_1R_2R_3R_4|$. Then $B=|R_1R_2(R_3+2R_1)R_4|$. Here $|B|=|A|$ because we know that if we add any scalar multiple of some row of an determinant to another row, the value of the determinant remains same.

Now $|CD|=|C||D|$ for any square matrices $C, D$ of same order. By this we have $$|2A^{-1}B|=2^4 |A^{-1}||B|=16|A|^{-1}|B|=16$$ since $|A|=|B|$.