Question

The value of the determinant \[\left| {\begin{array}{*{20}{c}} {{{\cos }^2}{{54}^ \circ }}&{{{\cos }^2}{{36}^ \circ }}&{\cot {{135}^ \circ }}\\ {{{\sin }^2}{{53}^ \circ }}&{\cot {{135}^ \circ }}&{{{\sin }^2}{{37}^ \circ }}\\ {\cot {{135}^ \circ }}&{{{\cos }^2}{{25}^ \circ }}&{{{\cos }^2}{{65}^ \circ }} \end{array}} \right|\]       is equal to

A. $$ - 2$$
B. $$ - 1$$
C. $$0$$  
D. $$1$$
Answer :   $$0$$
Solution :
Let, \[\Delta = \left| {\begin{array}{*{20}{c}} {{{\cos }^2}{{54}^ \circ }}&{{{\cos }^2}{{36}^ \circ }}&{\cot {{135}^ \circ }}\\ {{{\sin }^2}{{53}^ \circ }}&{\cot {{135}^ \circ }}&{{{\sin }^2}{{37}^ \circ }}\\ {\cot {{135}^ \circ }}&{{{\cos }^2}{{25}^ \circ }}&{{{\cos }^2}{{65}^ \circ }} \end{array}} \right|\]
\[ = \left| {\begin{array}{*{20}{c}} {{{\cos }^2}{{54}^ \circ }}&{{{\sin }^2}{{54}^ \circ }}&{ - 1}\\ {{{\cos }^2}{{37}^ \circ }}&{ - 1}&{{{\sin }^2}{{37}^ \circ }}\\ { - 1}&{{{\cos }^2}{{25}^ \circ }}&{{{\sin }^2}{{25}^ \circ }} \end{array}} \right|\]
$${C_1} \to {C_1} + {C_2} + {C_3}$$
\[\begin{array}{l} = \left| {\begin{array}{*{20}{c}} {{{\cos }^2}{{54}^ \circ } + {{\sin }^2}{{54}^ \circ } - 1}&{{{\sin }^2}{{54}^ \circ }}&{ - 1}\\ {{{\cos }^2}{{37}^ \circ } - 1 + {{\sin }^2}{{37}^ \circ }}&{ - 1}&{{{\sin }^2}{{37}^ \circ }}\\ { - 1 + {{\cos }^2}{{25}^ \circ } + {{\sin }^2}{{25}^ \circ }}&{{{\cos }^2}{{25}^ \circ }}&{{{\sin }^2}{{25}^ \circ }} \end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}} 0&{{{\sin }^2}{{54}^ \circ } - 1}\\ 0&{ - 1{{\sin }^2}{{37}^ \circ }}\\ 0&{{{\cos }^2}{{25}^ \circ }{{\sin }^2}{{25}^ \circ }} \end{array}} \right| = 0 \end{array}\]

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$  be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$  be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty
B. $$B$$  has as many elements as $$C$$
C. $$A = B \cup C$$
D. $$B$$  has twice as many elements as elements as $$C$$
View Answer
Releted Question 2

If $$\omega \left( { \ne 1} \right)$$  is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0
B. 1
C. $$i$$
D. $$\omega $$
View Answer
Releted Question 3

Let $$a, b, c$$  be the real numbers. Then following system of equations in $$x, y$$  and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$    $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$    $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$     has

A. no solution
B. unique solution
C. infinitely many solutions
D. finitely many solutions
View Answer
Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$
B. $$A + B = A - B$$
C. $$A - B = B - A$$
D. $$AB=BA$$
View Answer

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Matrices and Determinants

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