Properties of Determinants

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\end{array}\right]$. Such that $AB=B$ and $a+d=5050$. Find the value of $\left(ad-bc\right)$.

The value of given determinant is

$\left|\begin{array}{ccc}x+4& 2x& 2x\\ 2x& x+4& 2x\\ 2x& 2x& x+4\end{array}\right|$

The determinant $\left|\begin{array}{ccc}3a& -a+b& -a+c\\ -b+a& 3b& -b+c\\ -c+a& -c+b& 3c\end{array}\right|$ is equal to

If the area of a triangle is $4$ square units and its vertices are at $\left(-2,0\right),\left(0,4\right)$ and $\left(0,k\right)$, then the value of $k$ is equal to

The area of a triangle with vertices at $\left(2,-1\right), \left(0,-1\right)$ and $\left(2,0\right)$ is

Let $A$ and $B$ be $3\times 3$ matrices with det $A=2$ and det $B=-3.$ If $B^t$ is the transpose of $B,$ then $det\left(-2A{B}^3{\left(A^{-1}\right)}^6{B}^t\right)=$

If the determinant $\left|\begin{array}{ccc}x+1& x& x\\ x& x+1& x\\ x& x& x+1\end{array}\right|$ is equal to 0, then $x$ must be equal to

The area of a triangle whose vertices are at the points $(3,2),(-2,-3)$ and $(-1,-8)$ is

If $A$ is a $3\times 3$ matrix with $detA=1$ and if $B$ is the $3\times 3$ matrix whose first row is twice the first row of $A$, whose second row is same as the second row of $A$ and whose third row is the sum of the second row of $A$ and the third row of $A$, then $detB$ is equal to

The value of $x$ for which the matrix $\left[\begin{array}{ccc}8& x& 0\\ 4& 0& 2\\ 12& 6& 0\end{array}\right]$ is singular, is -

If $\omega$ is a complex cube root of unity, then the value of the determinant $\left|\begin{array}{ccc}1& \omega & \omega +1\\ \omega +1& 1& \omega \\ \omega & \omega +1& 1\end{array}\right|$ is