
If $D = \left| {\begin{array}{*{20}{c}} \alpha &\beta \\ \gamma &\delta \end{array}} \right|$, then \[\left| {\begin{array}{*{20}{c}} {2\alpha }&{2\beta } \\ {2\gamma }&{2\delta } \end{array}} \right|\] is equal to
Answer
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Hint: It is given in the question that if $D = \left| {\begin{array}{*{20}{c}}
Complete step-by-step answer:
Note: Students frequently get confused while attempting to distinguish between the properties of a matrix and a determinant. In a matrix we take n common from each element of the matrix. In determinants, we take n common values from each row or column.
\alpha &\beta \\
\gamma &\delta
\end{array}} \right|$
Then, what is the value of $\left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right|$ .
First, we will assume the $\alpha ,\beta ,\gamma ,\delta $ and put it in the $\left| {\begin{array}{*{20}{c}}
\alpha &\beta \\
\gamma &\delta
\end{array}} \right|$ , then after, we will put the value of $\alpha ,\beta ,\gamma ,\delta $ in the $\left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right|$ and finally, after solving further, we will get the answer.
It is given in the question that if $D = \left| {\begin{array}{*{20}{c}}
\alpha &\beta \\
\gamma &\delta
\end{array}} \right|$
Then, what is the value of $\left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right|$ .
Let us assume $\alpha = 1,\beta = 2,\gamma = 3,\delta = 4$ .
Now, put the value of $\alpha ,\beta ,\gamma ,\delta $ in the equation $D = \left| {\begin{array}{*{20}{c}}
\alpha &\beta \\
\gamma &\delta
\end{array}} \right|$ , we get,
$\therefore D = \left| {\begin{array}{*{20}{c}}
1&2 \\
3&4
\end{array}} \right|$ .
Similarly, Put the value of $\alpha ,\beta ,\gamma ,\delta $ in the $\left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right|$ , we get,
$ = \left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{2\left( 1 \right)}&{2\left( 2 \right)} \\
{2\left( 3 \right)}&{2\left( 4 \right)}
\end{array}} \right|$
$ = \left| {\begin{array}{*{20}{c}}
2&4 \\
6&8
\end{array}} \right|$
Now, take out 2 common from the row 1 of the determinant.
$ = 2\left| {\begin{array}{*{20}{c}}
1&2 \\
6&8
\end{array}} \right|$
Now, take out 2 from row 2 of the determinant.
$ = 2 \times 2\left| {\begin{array}{*{20}{c}}
1&2 \\
3&4
\end{array}} \right|$
$ = 4\left| {\begin{array}{*{20}{c}}
1&2 \\
3&4
\end{array}} \right|$
=4D
Therefore, the value of $\left| {\begin{array}{*{20}{c}}
{2\alpha }&{2\beta } \\
{2\gamma }&{2\delta }
\end{array}} \right| = 4D$ .
For example: in any matrix $A = \left[ {\begin{array}{*{20}{c}}
1&1 \\
1&1
\end{array}} \right]$ , $nA = \left[ {\begin{array}{*{20}{c}}
n&n \\
n&n
\end{array}} \right]$ , so $\left[ {nA} \right] = n\left[ A \right]$ and
In any determinant $B = \left| {\begin{array}{*{20}{c}}
1&1 \\
1&1
\end{array}} \right|$ , $nB = \left| {\begin{array}{*{20}{c}}
n&n \\
n&n
\end{array}} \right|$ , so $\left| {nB} \right| = {n^m}\left| B \right|$ , where m is the order of determinant B.
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