Multiplying a row or column of a square matrix by a constant multiplies the determinant of the matrix by that constant.
Proof
For a matrix
\[A\]
with elements \[(a_{ij})\]
the determinant is\[\left| A \right| = \sum_{\sigma} sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... a_{nk_n}\]
.where
\[sgn (\sigma )\]
is the parity of the \[k_1 , \: k_2 ,..., \: k_n \]
ie the number of transitions required to bring \[k_1 , \: k_2 ,..., \: k_n \]
to the order \[1, \: 2, \: ..., \: n \]
.If an even number of transitions is required then
\[sgn( \sigma )=1 \]
and if an odd number of transitions is required then \[sgn( \sigma )=-1 \]
.Suppose a row of
\[A\]
is multiplied by \[\alpha\]
to give a matrix \[A.\]
.Every term in the determinant now includes a factor
\[c\]
and looks like \[sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... ca_{jk_j} ... a_{nk_n}= c sgn ( \sigma ) a_{1k_1} a_{2 k_2} ... a_{nk_n}\]
.