If $A = [a_{ij}]_{3 \times 3} = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$,then $a_{11} A_{21} + a_{12} A_{22} + a_{13} A_{23}$ is equal to

Write the minors and cofactors of the elements of the following determinant: $\left|\begin{array}{rr}2 & -4 \\ 0 & 3\end{array}\right|$

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -3 \\ -1 & 2 & 3 \end{bmatrix}$,then find the value of $A_{31} + A_{32} + A_{33}$,where $A_{ij}$ denotes the cofactor of the element $a_{ij}$ of matrix $A$.

If the value of a third-order determinant is $11$,then the value of the square of the determinant formed by the cofactors will be:

In the determinant $\left| \begin{array}{ccc} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \end{array} \right|$,the ratio of the cofactor to its minor of the element $-3$ is

Let ${\Delta _1} = \begin{vmatrix} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{vmatrix}$ and ${\Delta _2} = \begin{vmatrix} {\alpha _1} & {\beta _1} & {\gamma _1} \\ {\alpha _2} & {\beta _2} & {\gamma _2} \\ {\alpha _3} & {\beta _3} & {\gamma _3} \end{vmatrix}$. Then ${\Delta _1} \times {\Delta _2}$ can be expressed as the sum of how many determinants?