If we change the one column of a determinant with greater values, what will be the effect on determinant. If $c_1 \geq d_1, c_2 \geq d_2$ and $c_3 \geq d_3$ and we change the third coulumn of $ \det\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\a_3 & b_3 & c_3 \end{pmatrix}$ as $ \det\begin{pmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\a_3 & b_3 & d_3 \end{pmatrix}$, what will be the relation between these determinants. My intuition is:
$$\det\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\a_3 & b_3 & c_3 \end{pmatrix} \geq \det\begin{pmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\a_3 & b_3 & d_3 \end{pmatrix}.$$
But I am not getting any reference for this.
 A: This is not true. Determinants are crazy, so changing some set of values results in weird changes overall. The determinant for the matrix in question is
$$D=c_1(a_2b_3-a_3b_2)-c_2(a_1b_3-a_3b_1)+c_3(a_1b_2-a_2b_1)$$
If we increase the $c_i$'s, this changes the determinant based on the determinants of the minors. In principle, the change in the determinant with respect to a particular element is equal in magnitude to the determinant of the element's corresponding minor, with a possible sign change based on the element's location.
A: This is false. Let the first matrix be
$\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0 \\
1 & 0 & 1
\end{bmatrix}
$ and the second
$\begin{bmatrix}
1 & 0 & 1\\
0 & 1 & 0 \\
1 & 0 & 1
\end{bmatrix}
$. Then $d_1=0$, $d_2=0$, $d_3=1$, $c_1=1$, $c_2=0$, and $c_3=1$. These obey your inequalities, but the first matrix has determinant 1 while the second has determinant 0.
Moreover, this isn't true even if you replace the $\leq$ with strict inequalities, as I can use the matrices $\begin{bmatrix}
1 & 2 & 0\\
0 & 2 & 0 \\
1 & 2 & 1
\end{bmatrix}
$ and
$\begin{bmatrix}
1 & 2 & 2\\
0 & 2 & 2 \\
1 & 2 & 2
\end{bmatrix}
$, the former of which has determinant 2 while the latter has determinant 0.
A: View the determinant as giving the signed volume of the parallelepiped
defined by the three vectors $\bf a, \bf b, \bf c$.
Then if $\bf c$ is in the same "direction" as  $\bf w = \bf a \times \bf b$, more rigorously
if $0 < \bf c \cdot \bf w$, and untill the increase $\Delta \bf c = \bf d - \bf c$ is in the same direction,
you obtain an increase of the (positive) volume.
Then it is easy to understand what happens in the four cases:
original volume = $\pm$, $\Delta \bf c \cdot \bf w = \pm$.
A: Suppose this was true. Then
$$det(a,b,c) \geq det(a,b,d)$$
$$-det(a,b,c) \leq -det(a,b,d)$$
$$det(-a,b,c) \leq det(-a,b,d)$$
provides a counterexample.
