In the previous section, we saw the third property of determinants. In this section, we will see the fourth property.
Property IV
This can be written in 8 steps:
1. Let Δ = $\left |\begin{array}{r}
a_1 &{ a_2 } &{ a_3 } \\
b_1 &{ b_2 } &{ b_3 } \\
c_1 &{ c_2 } &{ c_3 } \\
\end{array}\right | $.
2. We can write a new determinant Δ1 by multiplying any row (or column) by a constant k. For example, by multiplying the second row by k, we get:
Δ1 = $\left |\begin{array}{r}
a_1 &{ a_2 } &{ a_3 } \\
k b_1 &{ k b_2 } &{ k b_3 } \\
c_1 &{ c_2 } &{ c_3 } \\
\end{array}\right |$.
3. We have already evaluated Δ in the previous sections: Δ =
a1b2c3 − a1b3c2
− a2b1c3 + a2b3c1
+ a3b1c2 − a3b2c1
4. Let us evaluate Δ1. We will expand along R1.
5. In the same way, the reader may check the result by multiplying any column by k.
6. Based on the above steps, we can write:
If each element of a row (or any column) of a determinant is multiplied by a constant k, then the value of the determinant gets multiplied by k.
7. Suppose that, a row (or a column) is present in such a way that, the elements in that row (or column) has a common factor.
• Then we can take out that common factor. The elements will become smaller numerically, thereby making the calculations easier.
8. Now we will see an interesting case. It can be written in 5 steps:
(i) Let Δ = $\left |\begin{array}{r}
a_1 &{ a_2 } &{ a_3 } \\
b_1 &{ b_2 } &{ b_3 } \\
k a_1 &{ k a_2 } &{ k a_3 } \\
\end{array}\right | $.
• Here, R3 is kR1. That is., corresponding elements of R1 and R3 are proportional.
(ii) By applying property IV, we can write:
Δ = $ k \left |\begin{array}{r}
a_1 &{ a_2 } &{ a_3 } \\
b_1 &{ b_2 } &{ b_3 } \\
a_1 &{ a_2 } &{ a_3 } \\
\end{array}\right | $
(iii) But by applying property III, we can write:
$ \left |\begin{array}{r}
a_1 &{ a_2 } &{ a_3 } \\
b_1 &{ b_2 } &{ b_3 } \\
a_1 &{ a_2 } &{ a_3 } \\
\end{array}\right | ~=~0$
(iv) So from (ii), we get: Δ = k × 0 = 0
(v) We can write:
If corresponding elements of any two rows (or two columns) of a determinant are proportional (in the same ratio), then it's value is zero.
• In the above example, all ratios are same because:
$\frac{k a_1}{a_1}~=~\frac{k a_2}{a_2}~=~\frac{k a_3}{a_3}~=~k$
Now we have a clear understanding about property IV. Let us see an example. It can be written in 5 steps:
1. Let Δ = $\left |\begin{array}{r}
102 &{ 18 } &{ 36 } \\
1 &{ 3 } &{ 4 } \\
17 &{ 3 } &{ 6 } \\
\end{array}\right | $.
2. We can write:
Δ = $\left |\begin{array}{r}
6(17) &{ 6(3) } &{ 6(6) } \\
1 &{ 3 } &{ 4 } \\
17 &{ 3 } &{ 6 } \\
\end{array}\right | $.
3. By applying property IV, we get:
Δ = $6 \left |\begin{array}{r}
17 &{ 3 } &{ 6 } \\
1 &{ 3 } &{ 4 } \\
17 &{ 3 } &{ 6 } \\
\end{array}\right | $.
4. We see that, R1 and R3 are identical. So by applying property III, we get:
Δ = 6 × 0 = 0
In the next section, we will see Property V.
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