To solve this problem, we need to determine the number of 3-tuples \((α, β, γ)\) that satisfy the given equation. The equation involves determining the determinant of the adjugate matrix applied multiple times.
First, note the properties of the matrix:
To solve, we consider:
This indicates \(\det(A)=(2^8×3^4)(α-β)(β-γ)(γ-α)\).
Now, consider possible values for distinct tuples \( (α, β, γ) \):
Given no further constraints (e.g., values of \( α, β, γ \)), we count permutations that maintain distinctness:
Finally, verify the solution range:
Thus, the number of such 3-tuples \((α, β, γ)\) is \(42\).
Calculate the determinant of the matrix:

In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$) 
Method used for separation of mixture of products (B and C) obtained in the following reaction is: 
The numbers or functions that are kept in a matrix are termed the elements or the entries of the matrix.
The matrix acquired by interchanging the rows and columns of the parent matrix is termed the Transpose matrix. The definition of a transpose matrix goes as follows - “A Matrix which is devised by turning all the rows of a given matrix into columns and vice-versa.”