Determinants Te...

If $$\alpha, \beta$$ are the roots of $$x^2+x+1=0$$ then $$\begin{vmatrix} y+1 & \beta & \alpha\\ \beta & y+\alpha & 1\\ \alpha & 1 & y+\beta\end{vmatrix}=?$$

$$\begin{vmatrix} \sin ^{ 2 }{ \theta  }  & \cos ^{ 2 }{ \theta  }  \\ -\cos ^{ 2 }{ \theta  }  & \sin ^{ 2 }{ \theta  }  \end{vmatrix}=$$

The sum of the real roots of the equation
$$\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0$$ is equal to

$$\begin{vmatrix} a+ib & c+id \\ -c+id & a-ib \end{vmatrix}=$$?

For square matrices $$A$$ and $$B$$ of the same order, we have $$adj (AB) = ?$$

$$\begin{vmatrix} \cos { { 70 }^{ o } }  & \sin { { 20 }^{ o } }  \\ \sin { { 70 }^{ o } }  & \cos { { 20 }^{ o } }  \end{vmatrix}=$$?

Evaluate : $$\begin{vmatrix} \sin { { 23 }^{ o } }  & -\sin { { 7 }^{ o } }  \\ \cos { { 23 }^{ o } }  & \cos { { 7 }^{ o } }  \end{vmatrix}$$

If $$A = \begin{bmatrix} a& b\\c  & d\end{bmatrix}$$ then $$adj\ A = ?$$

If $$A$$ is a $$3-rowed$$ square matrix and $$|A| = 5$$ then $$|adj\ A| = ?$$

If $$|A| = 3$$ and $$A^{-1} = \begin{bmatrix}3 & -1\\ \dfrac {-5}{3} & \dfrac {2}{3}\end{bmatrix}$$ then $$adj\ A = ?$$