1 1 1
a b c
a^2 b^2 c^2
Factorise the determinant
To factorize the determinant, we need to determine if the given matrix has any common factors or if it can be rewritten in a simplified form.
In general, the determinant of a 3x3 matrix can be calculated using the following formula:
Det = a(ei - fh) - b(di - fg) + c(dh - eg)
In this case, we have the matrix:
1 1 1
a b c
a^2 b^2 c^2
The determinant can be expanded as follows:
Det = 1(b * c^2 - b^2 * c) - 1(a * c^2 - a^2 * c) + 1(a * b^2 - a^2 * b)
Simplifying further:
Det = bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b
Now we can factorize the expression. The common factor in the first three terms is 'c' and the common factor in the next three terms is 'b'. Therefore, we can write:
Det = c(bc - b^2 - ac) + b(ab - a^2)
Simplifying again:
Det = c(b(c - b) - a(c^2 - ab)) + b(a(b - a))
Finally, we can factor out a common factor of (c - b) from the first term and (b - a) from the second term:
Det = (c - b)(bc - ac - (c + b)(c - a)) + b(a - b)
Therefore, the factorization of the given determinant is:
Det = (c - b)(bc - ac - (c + b)(c - a)) + b(a - b)