Determinant of a Square Matrix

If $A$ is an orthogonal matrix of order $3$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$, then which of the following is/are correct $?$

$\mathrm{I}$. $\left|AB\right|=\pm 47$

$\mathrm{II}$. $AB=BA$

Select the correct answer using the code given below.

The matrix $A=\left[\begin{array}{ccc}1& 3& 2\\ 1& x-1& 1\\ 2& 7& x-3\end{array}\right]$ will have inverse for every real number $x$ except for

One of the roots of $\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|=0$ is

The roots of the equation $\left|\begin{array}{ccc}1& t-1& 1\\ t-1& 1& 1\\ 1& 1& t-1\end{array}\right|=0$ are

The value of the determinant $\left|\begin{array}{ccc}m& n& p\\ p& m& n\\ n& p& m\end{array}\right|$

What is the value of $\left|\begin{array}{ccc}-a^2& ab& ac\\ ab& -b^2& bc\\ ac& bc& -c^2\end{array}\right|$?

If $\mathrm{\Delta }=\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|$, then consider the following statements

$\mathrm{I}.$ If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0,$ then $\mathrm{\Delta }=abc$.

$\mathrm{II}$. If $a^{-1}+b^{-1}+c^{-1}=-1,$ then $\mathrm{\Delta }=0$.

Which of the above statement(s) is / are correct?

If the matrix $A=\left[\begin{array}{ccc}2-x& 1& 1\\ 1& 3-x& 0\\ -1& -3& -x\end{array}\right]$ is singular, then what is the solution set $S$$?$

What is the value of $\left|\begin{array}{cc}\cos 15°& \sin 15°\\ \cos 45°& \sin 45°\end{array}\right|\times \left|\begin{array}{cc}\cos 45°& \cos 15°\\ \sin 45°& \sin 15°\end{array}\right|$?

What is the value of $k$, if $\left|\begin{array}{lll}k& b+c& b^2+c^2\\ k& c+a& c^2+a^2\\ k& a+b& a^2+b^2\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(c-a\right)$?

If $5$ and $7$ are the roots of the equation $\left|\begin{array}{lll}x& 4& 5\\ 7& x& 7\\ 5& 8& x\end{array}\right|=0,$ then what is the third root$?$

The roots of the equation $\left|\begin{array}{lll}x& \alpha & 1\\ \beta & x& 1\\ \beta & \gamma & 1\end{array}\right|=0$ are independent of

If $A,B$ and $C$ are the angles of a triangle, then the value of $\Delta =\left|\begin{array}{ccc}-1& \cos C& \cos B\\ \cos C& -1& \cos A\\ \cos B& \cos A& -1\end{array}\right|$ is

If $p+q+r=a+b+c=0$, then the value of $\left|\begin{array}{lll}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ is

If $f\left(x\right)=\left|\begin{array}{lll}x& a& a\\ a& x& a\\ a& a& x\end{array}\right|=0,$ then

$\left|\begin{array}{lll}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right|+\left|\begin{array}{lll}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\right|+\dots \dots +\left|\begin{array}{ccc}n& 0& 0\\ 0& n+1& 0\\ 0& 0& n+2\end{array}\right|$ is equal to

The determinant $\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|$ is equal to zero, if

If $\left[.\right]$ denotes the greatest integer less than or equal to the real number under consideration and $-1\le x<0; 0\le y<1; 1\le z<2,$ then the value of the determinant $\left|\begin{array}{ccc}\left[x\right]+1& \left[y\right]& \left[z\right]\\ \left[x\right]& \left[y\right]+1& \left[z\right]\\ \left[x\right]& \left[y\right]& \left[z\right]+1\end{array}\right|$ is

The value of the determinant $\Delta =\left|\begin{array}{ccc}1!& 2!& 3!\\ 2!& 3!& 4!\\ 3!& 4!& 5!\end{array}\right|$ is

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