Advertisements
Advertisements
Question
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}, where A, B, C \text{ are the angles of }∆ ABC .\]
Solution
\[\begin{vmatrix}\sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1\end{vmatrix}\]
\[ = \begin{vmatrix}\sin^2 A - \sin^2 B & \cot A - \cot B & 0 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C - \sin^2 B & \cot C - \cot B & 0\end{vmatrix} \left[ \text{ Applying } R_1 \to R_1 - R_2 \text{ and }R_3 \to R_3 - R_2 \right]\]
\[ = \begin{vmatrix}\sin\left( A + B \right)\sin\left( A - B \right) & \frac{\cos A\sin B - \cos B\sin A}{\sin A\sin B} & 0 \\ \sin^2 B & \cot B & 1 \\ \sin\left( C + B \right)\sin\left( C - B \right) & \frac{\cos C\sin B - \cos B\sin C}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \begin{vmatrix}\sin\left( \pi - C \right)\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & cot B & 1 \\ \sin\left( \pi - A \right)\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix} \left[ \because A + B + C = \pi \right]\]
\[ = \begin{vmatrix}\sin C\sin\left( A - B \right) & \frac{- \sin\left( A - B \right)}{\sin A\sin B} & 0 \\ \sin^2 B & \frac{\cos B}{\sin B} & 1 \\ \sin A\sin\left( C - B \right) & \frac{- \sin\left( C - B \right)}{\sin B\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B}\begin{vmatrix}\sin C & \frac{- 1}{\sin A} & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A & \frac{- 1}{\sin C} & 0\end{vmatrix}\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}\sin C\sin A & - 1 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to \sin A R_1\text{ and }R_3 \to \sin C R_3 \right]\]
\[ = \frac{\sin\left( A - B \right)\sin\left( C - B \right)}{\sin B\sin A\sin C}\begin{vmatrix}0 & 0 & 0 \\ \sin^2 B & \cos B & 1 \\ \sin A\sin C & - 1 & 0\end{vmatrix} \left[ \text{ Applying }R_1 \to R_1 - R_3 \right]\]
\[ = 0\]
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Solve system of linear equations, using matrix method.
5x + 2y = 4
7x + 3y = 5
Solve the system of linear equations using the matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]
Prove the following identities:
\[\begin{vmatrix}x + \lambda & 2x & 2x \\ 2x & x + \lambda & 2x \\ 2x & 2x & x + \lambda\end{vmatrix} = \left( 5x + \lambda \right) \left( \lambda - x \right)^2\]
Solve the following determinant equation:
Solve the following determinant equation:
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Prove that :
Prove that :
Given: x + 2y = 1
3x + y = 4
x+ y = 5
y + z = 3
x + z = 4
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
State whether the matrix
\[\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}\] is singular or non-singular.
If \[A = \begin{bmatrix}0 & i \\ i & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\] , find the value of |A| + |B|.
If \[A = \left[ a_{ij} \right]\] is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.
Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].
Write the value of \[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix} .\]
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
If \[A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}\]. Write the cofactor of the element a32.
If \[\begin{vmatrix}3x & 7 \\ - 2 & 4\end{vmatrix} = \begin{vmatrix}8 & 7 \\ 6 & 4\end{vmatrix}\] , find the value of x.
If \[A = \begin{bmatrix}\cos\theta & \sin\theta \\ - \sin\theta & \cos\theta\end{bmatrix}\] , then for any natural number, find the value of Det(An).
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
Solve the following system of equations by matrix method:
3x + y = 19
3x − y = 23
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 3y = 5
6x + 9y = 15
Use product \[\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}\] to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0
If \[\begin{bmatrix}1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}\], find x, y and z.
Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.
`abs ((("b" + "c"^2), "a"^2, "bc"),(("c" + "a"^2), "b"^2, "ca"),(("a" + "b"^2), "c"^2, "ab")) =` ____________.
In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?
Let P = `[(-30, 20, 56),(90, 140, 112),(120, 60, 14)]` and A = `[(2, 7, ω^2),(-1, -ω, 1),(0, -ω, -ω + 1)]` where ω = `(-1 + isqrt(3))/2`, and I3 be the identity matrix of order 3. If the determinant of the matrix (P–1AP – I3)2 is αω2, then the value of α is equal to ______.
