What are the implications of the fact that the determinant of this general matrix is 4?

Question

Suppose we have the following matrix $A\in M(4\times 4;\;\mathbb{R})$ with $\det(A) =4$ : $$A= \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix} $$

What are the implications of the fact that the determinant of this general matrix is 4 or in other words, what are the characteristics/properties that follow from that fact?

EDIT:

How does this help me if (for example) I want to compute the determinant of \begin{pmatrix} 2a_{31}+a_{21} & 2a_{32}+a_{22} & 2a_{33}+a_{23} & 2a_{34}+a_{24} \\ a_{21} & a_{22} & a_{23} & a_{24}\\ 3a_{11} & 3a_{12} & 3a_{13} & 3a_{14}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}?

Answer 1

$$ \begin{vmatrix} 2a_{31}+a_{21} & 2a_{32}+a_{22} & 2a_{33}+a_{23} & 2a_{34}+a_{24} \\ a_{21} & a_{22} & a_{23} & a_{24}\\ 3a_{11} & 3a_{12} & 3a_{13} & 3a_{14}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} =\\ \begin{vmatrix} 2a_{31} & 2a_{32} & 2a_{33} & 2a_{34} \\ a_{21} & a_{22} & a_{23} & a_{24}\\ 3a_{11} & 3a_{12} & 3a_{13} & 3a_{14}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} = 2 \cdot 3 \begin{vmatrix} a_{31} & a_{32} & a_{33} & a_{34} \\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{11} & a_{12} & a_{13} & a_{14}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} = -2 \cdot 3 \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} $$

Answer 2

Let $ A\in M(n\times n;\;\mathbb{R})$ invertible and let $a_1,...,a_n$ be the columns of $A$.

Put $B=( \frac{4}{\det(A)}a_1,a_2,..,a_n)$, then we have $\det(B)=4.$