How Do Row Operations Affect Matrix Determinants?

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Homework Statement


If the determinant of a 3 x 3 matrix A is det(A) = 10, and the matrix B is obtained by multiplying the third row by 8, then det(B) = ___?

If the determinant of a 5 x 5 matrix A is det(A) = 9, and the matrix C is obtained from A by swapping the second and fourth rows, then det(C) =___?

Homework Equations



Ok, so do I use a generic matrix, or what? SO confused...

The Attempt at a Solution

 
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Hi LaraCroft! :smile:

Useful tip: write out a few matrices (easy ones, with lots of zeros :wink:), and see what happens …

then you can set about proving why :smile:
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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