Determinants

\(\left|\begin{array}{cc}\mathrm{x}+1 & \mathrm{x}-1 \\ \mathrm{x}^2+\mathrm{x}+1 & \mathrm{x}^2-\mathrm{x}+1\end{array}\right|\) is equal to :

If inverse of matrix \(\left[\begin{matrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{matrix}\right]\) is the matrix \(\left[\begin{matrix}1 & 3 & 3 \\ 1 & \lambda & 3 \\ 1 & 3 & 4\end{matrix}\right]\), then value of \(\lambda\) is:

If \(\left|\begin{array}{rrr}-a & b & c \\ a & -b & c \\ a & b & -c\end{array}\right|=k a b c\), then the value of \(k\) is :

If inverse of matrix \(\left[\begin{matrix}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{matrix}\right]\) is the matrix \(\left[\begin{matrix}1 & 3 & 3 \\ 1 & \lambda & 3 \\ 1 & 3 & 4\end{matrix}\right]\), then value of \(\lambda\) is:

If A is a square matrix of order 3 such that the value of \(|adj\cdot \mathrm{A}|=8\), then the value of \(\left|{\mathrm{A}}^{\mathrm{T}}\right|\) is :

If \(A\) is a square matrix of order \(3\) such that the value of \(|adj\cdot \mathrm{A}|=8\), then the value of \(\left|{\mathrm{A}}^{\mathrm{T}}\right|\) is :

If \(\left|\begin{array}{rrr}-a & b & c \\ a & -b & c \\ a & b & -c\end{array}\right|=k a b c\), then the value of \(k\) is :

\(\left|\begin{array}{cc}\mathrm{x}+1 & \mathrm{x}-1 \\ \mathrm{x}^2+\mathrm{x}+1 & \mathrm{x}^2-\mathrm{x}+1\end{array}\right|\) is equal to :

If A is an invertible matrix of order 2 , then \(\det \left({A}^{-1}\right)\) is equal to :

If A is a nonsingular square matrix of order \(3\times 3\) and \(|A|=2\), then \(|adjA|\) is equal to -

If A is a square matrix of order \(2\times 2\), then \(|5A|\) is equal to

Assertion (A) : For matrix \(\mathrm{A}=\left[\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right]\), where \(\theta \in[0,2 \pi]\), \(|\mathrm{A}| \in[2,4]\).
Reason \((R): \quad \cos \theta \in[-1,1], \forall \theta \in[0,2 \pi]\).

If \(\left|\begin{matrix}x & 2 \\ 18 & x\end{matrix}\right|=0\), then x is equal to