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Q.
If each element of a row (or a column) of a determinant is multiplied by a constant $k$, then its value gets...A... by $k$. Here, A refers to
Determinants
Solution:
If each element of a row (or a column) of determinant is multiplied by a constant $k$, then its value gets multiplied by $k$.
Verification Let $\Delta=\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$
and $\Delta_1$ be the determinant obtained by multiplying the elements of the first row by $k$.
Then, $\Delta_1=\begin{vmatrix}k a_1 & k b_1 & k c_1 \\a_2 & b_2 & c_2 \\a_3 & b_3 & c_3\end{vmatrix}$
By expanding along tirst row, we get
$ \Delta_1=k a_1\left(b_2 c_3-b_3 c_2\right)-k b_1\left(a_2 c_3-c_2 a_3\right)
+k c_1\left(a_2 b_3-b_2 a_3\right)$
$=k\left[a_1\left(b_2 c_3-b_3 c_2\right)-b_1\left(a_2 c_3-c_2 a_3\right)\right.
\left.+c_1\left(a_2 b_3-b_2 a_3\right)\right]$
Hence, $\begin{vmatrix}k a_1 & k b_1 & k c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}=k\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$
By this property, we can take out any common factor from anyone row or anyone column of a given determinant.