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.
