Higher Secondary Mathematics: 20.2

In the previous section, we saw how to obtain the determinant of order 3. In this section, we will see some solved examples.

Solved example 20.3
Evaluate the determinant $\Delta \;=\; \left|\begin{array}{r}                1    &{    2    }    &{    4    }    \\
-1    &{    3    }    &{    0    }    \\
4    &{    1    }    &{    0    }    \\
\end{array}\right|$
Solution:
We will expand along C₃ because, it has the maximum number of zeroes. We get:

Solved example 20.4
Evaluate $\Delta \;=\; \left|\begin{array}{r}                0    &{    \sin \alpha    }    &{    -\cos \alpha    }    \\
-\sin \alpha    &{    0    }    &{    \sin \beta    }    \\
\cos \alpha    &{    -\sin \beta    }    &{    0    }    \\
\end{array}\right|$
Solution:
Expanding along R₁, we get:

Solved example 20.5
Find the values of x for which $\left|\begin{array}{r}           3   &{   x   }   \\
x   &{   1   }   \\
\end{array}\right|~=~\left|\begin{array}{r}
3   &{    2    }   \\
4    &{   1   }   \\
\end{array}\right|$
Solution:

Solved example 20.6
Find the values of x if
(i) $\left|\begin{array}{r}           2    &{    4    }   \\
5    &{   1   }   \\
\end{array}\right|~=~\left|\begin{array}{r}
2x    &{    4    }   \\
6    &{    x    }   \\
\end{array}\right|$
(ii) $\left|\begin{array}{r}           2    &{    3    }   \\
4    &{    5    }   \\
\end{array}\right|~=~\left|\begin{array}{r}
x    &{    3    }   \\
2x    &{    5    }   \\
\end{array}\right|$
Solution:
Part (i):

Part (ii):

Solved example 20.7
If $\left|\begin{array}{r}           x    &{    2    }   \\
18    &{    x    }   \\
\end{array}\right|~=~\left|\begin{array}{r}
6    &{    2    }   \\
18    &{    6    }   \\
\end{array}\right|$, then x is equal to
(A) 6        (B) ∓ 6        (C) -6        (D) 0
Solution: