If a $4 \times 4$ matrix $A$ with rows $v_1, v_2, v_3$ and $v_4$ has determinant $\det A = 9$, then $$\det \begin{pmatrix}v_1\\ 3v_2+4v_3\\ 9v_2+3v_3\\ v_4\end{pmatrix} = \ ?$$
What I did was multiply the initial determinant by $3$ and then $3$ again as that is what the rows were being multiplied by on their own. The addition of other rows does not seem to change the determinant according to the rules. However, my answer of $81$ was incorrect.
Any help would be highly appreciated!
Best Answer
Hint: If your original matrix is $A$, the new one is $\pmatrix{1 & 0 & 0 & 0\cr 0 & 3 & 4 & 0\cr 0 & 9 & 3 & 0\cr 0 & 0 & 0 & 1\cr} A$.