If A + B + C = \(\pi \), then, the value of \(\left| {\begin{array}{*{20}{c}} {\sin \left( {A + B + C} \right)}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {\cos \left( {A + B} \right)}&{ - \tan A}&0 \end{array}} \right|\) is
0
Concept:
- sin π = 0 and cos (π - θ) = - cos θ
Calculation:
\(A = \left| {\begin{array}{*{20}{c}} {\sin \left( {A + B + C} \right)}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {\cos \left( {A + B} \right)}&{ - \tan A}&0 \end{array}} \right|\)
Put \(A + B + C = π \), and \(A + B = π - C\)
\(\left| {\begin{array}{*{20}{c}} {\sin \left( {π } \right)}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {\cos \left( {π-C} \right)}&{ - \tan A}&0 \end{array}} \right|\)
\(\left| {\begin{array}{*{20}{c}} {0}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {-cos \left( {C} \right)}&{ - \tan A}&0 \end{array}} \right|\)
A is a skew symmetric matrix.
∴ ⇒ |A| = 0
Important Questions from Evaluation of Determinants
The value of the determinant \(\left| {\begin{array}{*{20}{c}} {1 - {\rm{\alpha }}}&{{\rm{\alpha }} - {{\rm{\alpha }}^2}}&{{{\rm{\alpha }}^2}}\\ {1 - {\rm{\beta }}}&{{\rm{\beta }} - {{\rm{\beta }}^2}}&{{{\rm{\beta }}^2}}\\ {1 - {\rm{\gamma }}}&{{\rm{\gamma }} - {{\rm{\gamma }}^2}}&{{{\rm{\gamma }}^2}} \end{array}} \right|\) is equal to
The element in the i th row and the j th column of a determinant of third order is equal to 2(i + j). What is the value of the determinant?
If x, y, z are distinct real numbers and \(\left| {\begin{array}{*{20}{c}} x&{{x^2}}&{2 + {x^3}}\\ y&{{y^2}}&{2 + {y^3}}\\ z&{{z^2}}&{2 + {z^3}} \end{array}} \right| = 0\), then xyz =
Let p, q and r be three distinct positive real numbers. If \(\rm D = \left| {\begin{array}{*{20}{c}} \rm p&\rm q&\rm r\\ \rm q&\rm r&\rm p\\ \rm r&\rm p&\rm q \end{array}} \right|,\) then which one of the following is correct?
If a 1, a 2, a 3, _ _ _ _ _, a 9are in GP, then what is the value of the following determinant?
\(\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|\)
