Adjoint of a Matrix

For a matrix $A$, if $\left|A\right|=6$ and $\left(\mathrm{adj}\right)A=\left[\begin{array}{ccc}1& -2& 4\\ 4& 1& 1\\ -1& k& 0\end{array}\right]$, then $k$ is equal to

If $A=\left[\begin{array}{ccc}2& 52& 152\\ 4& 106& 358\\ 6& 162& 620\end{array}\right]$, then the determinant of the matrix $\mathrm{adj}\left(2A\right)$ is equal to

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$ , then $\mathrm{Adj}\left(3A^2+12A\right)$ is equal to:

If A is a $3\times 3$ matrix such that $\left|5 adjA\right|=5, then \left|A\right| is equal to:$

Let $A$ be a matrix of order $3\times 3$ such that $\det \left(\mathrm{A}\right)=2,B=2A^{-1}$ and $C=\frac{\left(\mathrm{adjA}\right)}{\sqrt[3]{16}}$ , then the value of $\det \left(A^3{B}^2{C}^3\right)$ is

Let $P$ be a matrix of order $3\times 3$ and $det\left(P\right)=3$, then the value of $det\left(adj\left(adj P\right)\right)$ is

For any $2\times 2$ matrix, if $A\left(adj A\right)=\left[\begin{array}{cc}20& 0\\ 0& 20\end{array}\right]$, then $\left|A\right|$ equals

If the value of a third order determinant is $10$, then the value of the square of the determinant formed by its cofactors will be

If $A=\left[\begin{array}{l}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}7\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\end{array}\right]$, then $adj\left(A\right)=$

If $A=\left[\begin{array}{rrr}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, then $adj\left(\mathrm{adj}A\right)$ is

If $P=\left[\begin{array}{lll}1& x& 0\\ 1& 3& 0\\ 2& 4& -2\end{array}\right]$ is the adjoint of $3\times 3$ matrix $A$ and $\left|A\right|=4,$ then $x$ is

If $A=\left[\begin{array}{ll}2& 0\\ 1& 5\end{array}\right]$ and $B=\left[\begin{array}{ll}1& 4\\ 2& 0\end{array}\right]$, then $\left|\mathrm{adj}\left(AB\right)\right|=$

If $|adj(adj\mathrm{A}\left)\right|=|\mathrm{A}{|}^9,$ then the order of the square matrix is

Let $A$ be a matrix of order $3\times 3$ such that $\det \left(A\right)=2, B=2A^{-1}$and $C=\frac{\left(adjA\right)}{\sqrt[3]{16}}$ , then the value of $\det \left(A^3{B}^2{C}^3\right)$ is

If the value of a third order determinant is $12$, then the value of the determinant formed by replacing each element by its co-factor will be 

The adjoint of the matrix $A=\left[\begin{array}{cc}2& -3\\ 3& 5\end{array}\right]$ is

The sum of the cofactors of the elements of second row of the matrix $\left[\begin{array}{ccc}1& 3& 2\\ -2& 0& 1\\ 5& 2& 1\end{array}\right]$ is

Let matrix $A=\left[\begin{array}{lll}x& y& 2\\ 1& 2& 3\\ 1& 1& 2\end{array}\right]$ where $x,y\in N.$ If $|adj(adj(adj(adjA\left)\right)\left)\right|=3^{32}·5^{16},$ then number of such
matrix $A$ is equal to

If $A=\left[\begin{array}{ccc}2& 3& 0\\ 1& {\sin }^{2}x& -{\cos }^{2}x\\ {\cos }^{2}x& {\cos }^{4}x& {\sin }^{4}x\end{array}\right]$ and $A\left(\mathrm{Adj} A\right)=K\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$, then the value of $K$ is

Let $A$ and $B$ be square matrices of order $3$ satisfying $A+adjA=B$ and $\left|A\right|=3$ then $\left|\left(adj A\right)B-3I\right|$ is equal to