The property states that,
"If to each element of any row (or column) of a matrix, product of a scalar and a corresponding element of any other row (or column) is added, the determinant of the new matrix is same as that of the original matrix"
First of all, I calculated the determinant of the original third-order non-zero matrix, say $A$.
Input: I first tested this property by adding the "product of a scalar $k_1$ and the corresponding elements of any column, say $C_2$" to only one column, say $C_1$, of the original matrix and then calculated the determinant of the new matrix $B$.
Result: $\det{B}=\det{A}$ (Property verified)
Input: Then, I tested this property by adding the "product of a scalar $k_1$ and the corresponding elements of column $C_2$" to $C_1$ and by adding the "product of a different scalar $k_2$ and the corresponding elements of another column $C_3$ to the column $C_2$ as well. A new matrix appeared, say $C$.
Result: $\det {C} = \det {A}$ (Property verified)
Input: Then, ultimately, I tested this property by taking matrix $C$ and adding the "product of a new scalar $k_3$ and the corresponding elements of the column $C_1$" to column $C_3$. A new matrix $D$ came up.
Result: $\det {D}= (\det {A})(1+k_1k_2k_3)$
$ \implies \det {D} \neq \det {A}$, unless $k_1=0$ or $k_2=0$ or $k_3=0$ or $k_1=k_2=k_3=0$ (Property invalid)
Note: In first two inputs, changes are made with matrix A directly however in third input changes are made to C in order to save time.
I think, my understanding to this property is flawed. Ensure me if it is the case.
The problem is in your incorrect applying the property for Input 3. You take the matrix with columns $C_k$ $$ A=\left[\matrix{C_1 & C_2 & C_3}\right]. $$
The correct way to apply the property in Case 3 is $$ D_{\text{ok}}=\left[\matrix{\color{magenta}{C_1+k_1C_2} & C_2+k_2C_3 & C_3+k_3\color{magenta}{(C_1+k_1C_2)}}\right] $$