2 Questions concerning determinants

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SUMMARY

The discussion centers on two mathematical questions regarding determinants and permutations. The first question asks for the number of even permutations in a 5x5 matrix, which relates to the concept of factorials in combinatorial mathematics. The second question involves calculating the area of a triangle formed by two vectors, v = (3,2) and w = (4,1), using determinants. The area can be derived from the determinant of a matrix formed by these vectors, specifically through the vector cross product method.

PREREQUISITES
  • Understanding of permutations and their properties, specifically even and odd permutations.
  • Familiarity with factorials and their application in combinatorial problems.
  • Knowledge of determinants and their calculation for 2x2 matrices.
  • Concept of vector cross products and their geometric interpretation in relation to area.
NEXT STEPS
  • Study the properties of permutations, focusing on even and odd classifications.
  • Learn how to calculate determinants for both 2x2 and 3x3 matrices.
  • Explore the relationship between vector cross products and area calculations in geometry.
  • Investigate advanced combinatorial techniques involving factorials and permutations.
USEFUL FOR

Students in mathematics, particularly those studying linear algebra, combinatorics, or geometry, will benefit from this discussion. It is also useful for educators looking to enhance their teaching of determinants and permutations.

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Homework Statement



First Question. How many even Permutations does a 5x5 matrix have? In other words how many permutations are there that would make it +1 instead of -1.

Second Question. v= (3,2) w= (4,1) use determinants to find the area of a triangle with sides v, w and v+w

Homework Equations





The Attempt at a Solution



First Question. I know you can write them all out, but is there some kind of formula to find how many permutations there are and determine whether the are even or odd? I assume it has something to do with factorials

Second question. I don't know what to do with the matrix when it is not a square, so how do you find a determinant of a triangle? I could use 1/2bh but the problem asks to use determinants specifically
 
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first - how many choices is there for the first row? then say you have chosen first row, how many for the 2nd... and so on

2nd - do you know about vector cross products? the length of a vector cross product is the area of the parallelogram made by the 2 vectors... relate this to the determinant equation
 

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