Proving that there is no invertible matrix with zero row sums using determinants

Question

I have the following question which I know I should use the determinant to solve. Here it is:

Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} a_{11}+a_{12}+a_{13}&=0 \\ a_{21}+a_{22}+a_{23}&=0 \\ a_{31}+a_{32}+a_{33}&=0 \end{align*}$$

I know that the answer is no, that there is no such matrix, but I’d like to know how to solve this question with determinants.

Answer 1

The determinant is invariant under adding single multiples of one column to another. So using this we can add Column $2$ to Column $1$. Then add Column $3$ to Column $1$. This results in a column of zeroes. Hence, the determinant of $A$ is $0$ so it cannot be invertible.