Minor and Cofactors

If $A_{ij}$ and $M_{ij}$ are the cofactors and minors of $△=\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|,$ then

If ${△}_1=\left|\begin{array}{ccc}{y}^5{z}^6\left(z^3-y^3\right)& x^4{z}^6\left(x^3-z^3\right)& x^4{y}^5\left(y^3-x^3\right)\\ y^2{z}^3\left(y^6-z^6\right)& x{z}^3\left(z^6-x^6\right)& x{y}^2\left(x^6-y^6\right)\\ y^2{z}^3\left(z^3-y^3\right)& x{z}^3\left(x^3-z^3\right)& x{y}^2\left(y^3-x^3\right)\end{array}\right|$ and ${△}_2=\left|\begin{array}{ccc}x& y^2& z^3\\ x^4& y^5& z^6\\ x^7& y^8& z^9\end{array}\right|$, then the value of ${△}_1{△}_2$ is 

For $\left|\begin{array}{ccc}2& 3& 5\\ 1& 0& 7\\ -1& -2& 4\end{array}\right|$, the sum of minor and cofactor of $7=$ _____

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

Find the cofactor of an element $a_{33}$ for the following determinant $\left|\begin{array}{ccc}2& 4& 5\\ 7& 1& 4\\ 1& 5& 8\end{array}\right|$.

Find the minor of an element $a_{33}$ for the following determinant $\left|\begin{array}{ccc}3& -3& 5\\ 6& 1& 4\\ 1& 5& -7\end{array}\right|$.

Find the minor of an element $a_{11}$ for the following determinant $\left|\begin{array}{ccc}3& -3& 5\\ 6& 1& 4\\ 1& 5& -7\end{array}\right|$.

Find the cofactor of an element $a_{12}$ for the following determinant $\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$.

Find the minor of an element $a_{12}$ for the following determinant $\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right|$.

If $A=\left[\begin{array}{ccc}1& 1& 0\\ 2& 1& 5\\ 1& 2& 1\end{array}\right],$ then $a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}=$ _______

The matrix of co-factors for the matrix $\left[\begin{array}{ll}1& 3\\ 4& -1\end{array}\right]$ is

Find the adjoint of the following matrix $\left[\begin{array}{ccc}1& -1& 2\\ -2& 3& 5\\ -2& 0& -1\end{array}\right]$

Find the matrix of cofactors for the following matrix $\left[\begin{array}{cc}1& 3\\ 4& -1\end{array}\right]$

Find the co-factors of the elements of the following matrix $\left[\begin{array}{ccc}1& -1& 2\\ -2& 3& 5\\ -2& 0& -1\end{array}\right]$

Find the co-factors of the elements of the following matrix $\left[\begin{array}{cc}-1& 2\\ -3& 4\end{array}\right]$

Let $A=\left[\begin{array}{cc}x& 2y\\ -1& y\end{array}\right]; x, y\in \mathit{R}$. If $A{A}^T=\left[\begin{array}{cc}1& 0\\ 0& \alpha \end{array}\right]\mathbf{(}\mathit{\alpha }\mathbf{\in }\mathit{R}\mathbf{)},$ then $\alpha +y^2$ is equal to

Find the minor of the elements of second row of determinant $\left|\begin{array}{ccc}2& 3& 4\\ 3& 6& 5\\ 1& 8& 9\end{array}\right|$.

If the capital letters denotes the cofactors of the corresponding small letters in the determinant

$\mathrm{\Delta }=\mathrm{ }\left|\mathrm{ }\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\mathrm{ }\right|$ & $\mathrm{\Delta }\mathrm{\prime }=\mathrm{ }\left|\mathrm{ }\begin{array}{lll}{\mathrm{A}}_1& {\mathrm{B}}_1& {\mathrm{C}}_1\\ {\mathrm{A}}_2& {\mathrm{B}}_2& {\mathrm{C}}_2\\ {\mathrm{A}}_3& {\mathrm{B}}_3& {\mathrm{C}}_3\end{array}\mathrm{ }\right|$ then $\mathrm{\Delta }\mathrm{\Delta }\mathrm{\prime }$ equals -

If $\mathrm{\Delta }=\mathrm{ }\left|\mathrm{ }\begin{array}{lll}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\\ {\mathrm{c}}_1& {\mathrm{c}}_2& {\mathrm{c}}_3\end{array}\mathrm{ }\right|$, then value of ${\mathrm{b}}_1{\mathrm{B}}_1\mathrm{ }+\mathrm{ }{\mathrm{b}}_2{\mathrm{B}}_2\mathrm{ }+\mathrm{ }{\mathrm{b}}_3{\mathrm{B}}_3$ is where ${\mathrm{B}}_1,\mathrm{ }{\mathrm{B}}_2,\mathrm{ }{\mathrm{B}}_3$ are co-factor of ${\mathrm{a}}_1,\mathrm{ }{\mathrm{a}}_2,\mathrm{ }{\mathrm{a}}_3$ respectively-