Problem on determinants of 2x2 matrices

Show that if the rows of a $2 \times 2$ matrix are multiples of each other, then the determinant of the matrix is zero.
• Solution

Consider a $2 \times 2$ matrix where the rows are multiples of each other. That is, for some $k$, $$A = \begin{pmatrix} a & b \\ k a & kb\end{pmatrix}.$$
Recall that
Hence, \begin{align} \text{det } A = a (kb) - b (ka) = kab - kab = 0. \end{align}