If there are two identical rows or columns identical in a determinant, the value is 0. Is the converse true?

Question

I know that
$$A= \begin {vmatrix} q&a&a \\ z&b&b \\ x&c&c \\ \end {vmatrix} =0 $$ Because in determinants, if we have to two identical rows or columns of $A$ that are identical (in this case the second and the third columns are identical), then the value of $\det A = 0$.

But is the the converse true? i.e. If I have a determinant like so $$ \begin {vmatrix} q&a&a \\ z&b&b \\ x&c&v \\ \end {vmatrix} =0 $$ does that imply that $c=v$ (so as to make the rows identical as the other two elements are)?

Answer 1

No, it doesn't.

For instance, $$ \left|\begin{matrix} 1&0&0\\ 2&0&0\\ 3&1&2 \end{matrix}\right|=0 $$ In general, a zero determinant is equivalent to the columns (or rows) being linearly dependent. Two of them being equal is a special case of linear dependence.

Answer 2

$$\begin {vmatrix} q&a&a \\ z&b&b \\ x&c&v \\ \end {vmatrix} \xrightarrow{C_3\to C_3-C_2} \begin {vmatrix} q&a&0 \\ z&b&0\\ x&c&v-c\\ \end {vmatrix} =(v-c)(qb-az)=0\implies v=c\vee qb=az$$which means the converse is not true in general.