Why adding a row with another row in square matrix A doesn't change the $\det(A)$ value?

Question

Why adding a row with another multiplied row in square matrix $A$ doesn't change the $\det(A)$ value?

Answer 1

Because of:

1. the linearity of the determinant according to the lines
2. the anti-symmetry of the determinant.

Answer 2

To develop the answer of @mookid note that: multilinearity means that that the detrminat is a linear function of his (vector) rows $\mathbf{a}_h$, so you have: $$ \det (\mathbf{a}_1,\cdots,\mathbf{a}_i+k\mathbf{a}_{j},\cdots, \mathbf{a}_{j},\cdots \mathbf{a}_n)^T=\det(\mathbf{a}_1,\cdots,\mathbf{a}_i,\cdots \mathbf{a}_{j},\cdots \mathbf{a}_n)^T +k\det(\mathbf{a}_1,\mathbf{a}_{j},\cdots \mathbf{a}_{j},\cdots \mathbf{a}_n)^T $$ Now, the determinat is also antisymmetric and this means that if two rows are the same the determinat is null, so: $\det(\mathbf{a}_1,\mathbf{a}_{j},\cdots \mathbf{a}_{j},\cdots \mathbf{a}_n)^T=0$.