Question

`| ( 1, 1, 1, 1, 1, 1, 1), ( 2^6, 2^5, 2^4, 2^3, 2^2, 2, 1 ), ( 3^6, 3^5, 3^4, 3^3, 3^2, 3, 1 ), ( 4^6, 4^5, 4^4, 4^3, 4^2, 4, 1 ), ( 5^6, 5^5, 5^4, 5^3, 5^2, 5, 1 ), ( 6^6, 6^5, 6^4, 6^3, 6^2, 6, 1 ), ( 7^6, 7^5, 7^4, 7^3, 7^2, 7, 1 ) | =`?

Answer 1

`-24883200`

Explanation:

`"This is the determinant of a Vandermonde matrix."`
`"It is known that the determinant is then a product of the"`
`"differences of the base numbers (that or taken to successive"` `"powers)."`

`"So here we have "`
`(6!)(5!)(4!)(3!)(2!)`
`"= 24,883,200"`

`"There is one difference though with the Vandermonde matrix"`
`"and that is that the lowest powers are normally on the left side"`
`"of the matrix so the columns are mirrored, this gives an extra"`
`"minus sign to the result : "`

`"determinant = -24,883,200"`