Verify Property $1$ for $\Delta=\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$
Verify Property $1$ for $\Delta=\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$
Solution Expanding the determinant along first row, we have
$\Delta = 2\left| {\begin{array}{*{20}{r}}
0&4 \\
5&{ - 7}
\end{array}} \right| - ( - 3)\left| {\begin{array}{*{20}{r}}
6&4 \\
1&{ - 7}
\end{array}} \right| + 5\left| {\begin{array}{*{20}{r}}
6&0 \\
1&5
\end{array}} \right|$
$ = 2(0 - 20) + 3( - 42 - 4) + 5(30 - 0)$
$ = - 40 - 138 + 150 = - 28$
By interchanging rows and columns, we get
$\Delta_{1}=\left|\begin{array}{rrr}2 & 6 & 1 \\ -3 & 0 & 5 \\ 5 & 4 & -7\end{array}\right| \quad$ (Expanding along first column)
$=2\left|\begin{array}{rr}
0 & 5 \\
4 & -7
\end{array}\right|-(-3)\left|\begin{array}{rr}
6 & 1 \\
4 & -7
\end{array}\right|+5\left|\begin{array}{ll}
6 & 1 \\
0 & 5
\end{array}\right|$
${ = 2(0 - 20) + 3( - 42 - 4) + 5(30 - 0)}$
${ = - 40 - 138 + 150 = - 28}$
Clearly $\Delta=\Delta_{1}$
Hence, Property $1$ is verified.
Similar Questions
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- [JEE MAIN 2021]
The determinant $\left| {\begin{array}{*{20}{c}}{^x{C_1}}&{^x{C_2}}&{^x{C_3}}\\ {^y{C_1}}&{^y{C_2}}&{^y{C_3}}\\{^z{C_1}}&{^z{C_2}}&{^z{C_3}}\end{array}} \right|$ $=$
The determinant $\left| {\begin{array}{*{20}{c}}{^x{C_1}}&{^x{C_2}}&{^x{C_3}}\\ {^y{C_1}}&{^y{C_2}}&{^y{C_3}}\\{^z{C_1}}&{^z{C_2}}&{^z{C_3}}\end{array}} \right|$ $=$