SOLUTION: Hello! I need a lot of help here. I have a question that need to be proved but I don't know how to do. A=([1,0,0,0,0],[a,-2,0,0,0],[a^2,b,-3,0,0],[a^3,b^2,c,-4,0],[a^4,b^3,c

Question 852815: Hello!
I need a lot of help here. I have a question that need to be proved but I don't know how to do.
A=([1,0,0,0,0],[a,-2,0,0,0],[a^2,b,-3,0,0],[a^3,b^2,c,-4,0],[a^4,b^3,c^2,d,-5])
there are four answers provided.
a)|A|=5!
b)|A|=(-1)^5(1.2.3.4.5)
c)|A|=0 since the rows are related
d)It is not possible to calculate the determinant of A because we don't know the value of a,b,c and d.
Thank you! Your help means a world to me!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
From this page, http://www.math.lsa.umich.edu/~hochster/419/det.html, it states that

Fact 7. The determinant of a lower triangular matrix (or an upper triangular matrix) is the product of the diagonal entries. In particular, the determinant of a diagonal matrix is the product of the diagonal entries.

Since matrix A=([1,0,0,0,0],[a,-2,0,0,0],[a^2,b,-3,0,0],[a^3,b^2,c,-4,0],[a^4,b^3,c^2,d,-5]), which really looks like this

[1,0,0,0,0]
[a,-2,0,0,0]
[a^2,b,-3,0,0]
[a^3,b^2,c,-4,0]
[a^4,b^3,c^2,d,-5]

is lower triangular, this means that we just multiply the diagonal entries, which are: 1, -2, -3, -4, -5

So multiplying them gives us: 1*(-2)*(-3)*(-4)*(-5) = 120

which is equivalent to 5! since 5! = 5*4*3*2*1 = 120

The answer is a)|A|=5!
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