Determinant of a Matrix

The value of the determinant $\left|\begin{array}{ccc}1& \cos \left(\alpha -\beta \right)& \cos \alpha \\ \cos \left(\alpha -\beta \right)& 1& \cos \beta \\ \cos \alpha & \cos \beta & 1\end{array}\right|$ is

If $A, B$ and $C$ are the angles of a non-right angled triangle $ABC$, then the value of $\left|\begin{array}{ccc}\tan A& 1& 1\\ 1& \tan B& 1\\ 1& 1& \tan C\end{array}\right|$ is equal to

Which of the following is correct?

The value of the determinant $\left|\begin{array}{ccc}265& 240& 219\\ 240& 225& 198\\ 219& 198& 181\end{array}\right|$ is

The digits $A, B$ and $C$ are such that the three digit numbers $A88, 6B8, 86C$ are divisible by $72$, then the determinant $\left|\begin{array}{lll}A& 6& 8\\ 8& B& 6\\ 8& 8& C\end{array}\right|$ is divisible by

If $∆\left(x\right)=\left|\begin{array}{ccc}1& \cos x& 1-\cos x\\ 1+\sin x& \cos x& 1+\sin x-\cos x\\ \sin x& \sin x& 1\end{array}\right|$, then ${\int }_0^{\pi /4}∆\left(x\right)dx$ is equal to

If $A=\left[\begin{array}{ll}3& 4\\ 5& 7\end{array}\right],$ then $A·\left(\mathrm{adj} A\right)$ is equal to

If $A=\left[\begin{array}{lll}2& 0& 0\\ 2& 2& 0\\ 2& 2& 2\end{array}\right]$, then $\mathrm{adj}\left(\mathrm{adj}A\right)$ is equal to 

If $a, b$ and $c$ are in $\mathrm{AP}$, then determinant $\left|\begin{array}{lll}x+2& x+3& x+2a\\ x+3& x+4& x+2b\\ x+4& x+5& x+2c\end{array}\right|$ is

If $A=\left[\begin{array}{cc}1& \tan x\\ -\tan x& 1\end{array}\right],$ then the value of $\left|A^{\text{'}}{A}^{-1}\right|$ is

Let $A={\left[a_{ij}\right]}_{n\times n}$ be a square matrix and let $c_{ij}$ be cofactor of $a_{ij}$ in $A$. If $C=\left[c_{ij}\right],$ then

The sum of the products of the elements of any row of a determinant $A$ with the corresponding cofactors of the same row is always equal to

If $\alpha , \beta$ and $\gamma$ are the roots of $x^3+a{x}^2+b=0,$ then the value of $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{array}\right|$ is

The cofactor element of $0$ in determinant $\left|\begin{array}{rrr}-1& 2& 1\\ -2& 3& -3\\ 4& 0& -4\end{array}\right|$ is

The value of $\left|\begin{array}{cc}1+\cos \theta & \sin \theta \\ \sin \theta & 1-\cos \theta \end{array}\right|$ is