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# Equality of determinants

Let and suppose that is nilpotent. Show that if   commute then

Solution

Since is algebraically closed and commute this means that are simultaneously triangularizable,  there exists an invertible element such that both and are triangular. Since is both nilpotent and triangular, all its diagonal entries are zero and so the diagonal entries of are the same as the diagonal entries of . Thus,

because are both triangular and the determinant of a triangular matrix is the product of its diagonal entries. So,

The exercise along its solution have been migrated from here.

## 1 Comment

1. Note: If the matrices do not commute , the above result does not necessarily hold. For example:

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