Question

|101 102 103|
|106 107 108| solve the determinant | 1 2 3 |by using properties
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Answer 1

$\textbf{Given:}$

$\left|\begin{array}{ccc}101&102&103\\106&107&108\\1&2&3\end{array}\right|$

$\textbf{To find:}$

$\text{The value of the given detereminant}$

$\textbf{Solution:}$

$\text{Consider,}$

$\left|\begin{array}{ccc}101&102&103\\106&107&108\\1&2&3\end{array}\right|$

$=\left|\begin{array}{ccc}101&102&103\\5&5&5\\1&2&3\end{array}\right|$ $R_2\implies\,R_2-R_1$

$\text{Take 5 common from R_2 }$

$=5\left|\begin{array}{ccc}101&102&103\\1&1&1\\1&2&3\end{array}\right|$

$=5\left|\begin{array}{ccc}100&100&100\\1&1&1\\1&2&3\end{array}\right|$ $R_1\implies\,R_1-R_3$

$\text{Take 100 common from R_1 }$

$=500\left|\begin{array}{ccc}1&1&1\\1&1&1\\1&2&3\end{array}\right|$ $R_1\implies\,R_1-R_3$

$=0$

$(\because\textbf{If any two rows (or) columns of a determinant are equal,}$

$\textbf{then the value of the determinant is zero})$

$\therefore\bf\left|\begin{array}{ccc}101&102&103\\106&107&108\\1&2&3\end{array}\right|=0$

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