Question

Evaluate the value of $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ .

Answer 1

Hint: Here, we have to find the value of $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ .
To find the value of any determinant of order 3, say $A = \left| {\begin{array}{*{20}{c}}
  a&b&c \\
  d&e&f \\
  g&h&i
\end{array}} \right|$ , we do $A = a \cdot \left| {\begin{array}{*{20}{c}}
  e&f \\
  h&i
\end{array}} \right| - b \cdot \left| {\begin{array}{*{20}{c}}
  d&f \\
  g&i
\end{array}} \right| + c \cdot \left| {\begin{array}{*{20}{c}}
  d&e \\
  g&h
\end{array}} \right|$ .
So, using the above method we need to evaluate $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ .

Complete step-by-step answer:
Here, we are asked to find the value of $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ .
Let $A = \left| {\begin{array}{*{20}{c}}
  a&b&c \\
  d&e&f \\
  g&h&i
\end{array}} \right|$ be any determinant of order 3.
Now, to find the value of determinant of order 3, we do as $A = a \cdot \left| {\begin{array}{*{20}{c}}
  e&f \\
  h&i
\end{array}} \right| - b \cdot \left| {\begin{array}{*{20}{c}}
  d&f \\
  g&i
\end{array}} \right| + c \cdot \left| {\begin{array}{*{20}{c}}
  d&e \\
  g&h
\end{array}} \right|$ .
Similarly, we will find the value of the determinant $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ as $\Delta = 0 \cdot \left| {\begin{array}{*{20}{c}}
  0&{\sin \beta } \\
  { - \sin \beta }&0
\end{array}} \right| - \sin \alpha \cdot \left| {\begin{array}{*{20}{c}}
  { - \sin \alpha }&{\sin \beta } \\
  {\cos \alpha }&0
\end{array}} \right| + \left( { - \cos \alpha } \right) \cdot \left| {\begin{array}{*{20}{c}}
  { - \sin \alpha }&0 \\
  {\cos \alpha }&{ - \sin \beta }
\end{array}} \right|$
 $
   \Rightarrow \Delta = 0\left[ {0 - \left( {\sin \beta } \right)\left( { - \sin \beta } \right)} \right] - \sin \alpha \left[ {0 - \left( {\cos \alpha } \right)\left( {\sin \beta } \right)} \right] - \cos \alpha \left[ {\left( { - \sin \alpha } \right)\left( { - \sin \beta } \right) - 0} \right] \\
   \Rightarrow \Delta = 0 - \sin \alpha \left( { - \cos \alpha \sin \beta } \right) - \cos \alpha \left[ {\sin \alpha \sin \beta } \right] \\
   \Rightarrow \Delta = \sin \alpha \sin \beta \cos \alpha - \sin \alpha \sin \beta \cos \alpha \\
   \Rightarrow \Delta = 0 \\
 $

Thus, the value of $\Delta = 0$ .

Note: Here, the direct approach to evaluate the value of $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ , can be as:
If the given matrix or determinant is skew-symmetric, then the value of its determinant is always 0 if the order of matrix or determinant is an odd number.
Here, the given determinant is $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ .
The given determinant is skew-symmetric and order is 3, which is an odd number.
So, the value of $\Delta = \left| {\begin{array}{*{20}{c}}
  0&{\sin \alpha }&{ - \cos \alpha } \\
  { - \sin \alpha }&0&{\sin \beta } \\
  {\cos \alpha }&{ - \sin \beta }&0
\end{array}} \right|$ directly becomes 0.
Primary diagonal of matrix or determinant:
The elements in the form of ${a_{ij}},i = j$ , form a diagonal of any square matrix or determinant. That diagonal is called the primary diagonal of the given matrix or determinant.
Skew-symmetric:
The matrix given in the form of $\left[ {\begin{array}{*{20}{c}}
  0&a&{ - b} \\
  { - a}&0&c \\
  b&{ - c}&0
\end{array}} \right]$ is called a skew-symmetric matrix.