Question

​Solve the following determinant equation:

\[\begin{vmatrix}3 & - 2 & \sin\left( 3\theta \right) \\ - 7 & 8 & \cos\left( 2\theta \right) \\ - 11 & 14 & 2\end{vmatrix} = 0\]

 

Answer 1

Let Δ `=|(3,-2,sin(3theta)),(-7,8,cos(2theta)),(-11,14,2)|=0`

`=>|(1,-2,sin(3theta)),(1,8,cos(2theta)),(3,14,2)|=0`      `["Applying"  C_1->C_1+C_2]`

`=>|(1,-2,sin(3theta)),(0,10,cos(2theta)-sin(3theta)),(0,20,2-3sin(3theta))|=0`    `["Applying"  R_2->R_2-R_1  "and"  R_3->R_3-3R_1]`

`=>10(2-3sin(3theta))-20(cos(2theta))-sin(3theta)=0`

`=>20-10sin(3theta)-20cos(2theta)=0`

`=>sin(3theta)+2cos(2theta)-2=0`

`=>3sintheta-4sin^3theta+2-4sin^2theta-2=0`

`=>-sintheta(4sin^2theta+4sintheta-3)=0`

`=>sintheta=0    or    4sin^2theta+4sintheta-3=0`

`=>theta=npi     or   (2sintheta+3)(2sintheta-1)=0`

`=>theta=npi     or   sintheta=-3/2   or   sintheta=1/2`

`=>theta=npi    or    theta=npi+(-1)^npi/6,ninZ`