Linear Algebra and Determinant

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SUMMARY

The discussion focuses on constructing 4x4 matrices to illustrate the differences in calculating determinants using cofactor expansion versus elementary row operations. A matrix with non-zero diagonal elements sampled from the uniform distribution U(0,1) is proposed for easy computation via cofactor expansion. Conversely, a matrix with two identical rows, also sampled from U(0,1), is suggested for easy evaluation using row operations, which would yield a determinant of zero. The uniform distribution U(0,1) is defined as the distribution of values between 0 and 1.

PREREQUISITES
  • Understanding of matrix theory and determinants
  • Familiarity with cofactor expansion method for determinants
  • Knowledge of elementary row operations in linear algebra
  • Concept of uniform distribution, specifically U(0,1)
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn advanced techniques for calculating determinants, including cofactor expansion
  • Explore the implications of row operations on matrix determinants
  • Investigate the characteristics of the uniform distribution U(0,1) in statistical contexts
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and statisticians interested in matrix theory and random sampling techniques.

Swati
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1(a) Construct a 4*4 matrix whose determinant is easy to compute using cofactor expansion but hard to evaluate using elementary row operations.

(b) Construct a 4*4 matrix whose determinant is easy to compute using elementary row operations but hard to evaluate using cofactor expansion
 
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Swati said:
1(a) Construct a 4*4 matrix whose determinant is easy to compute using cofactor expansion but hard to evaluate using elementary row operations.

Would a matrix with all elements zero except for those on the diagonal which are a random sample of size 4 from U(0,1) qualify?

(b) Construct a 4*4 matrix whose determinant is easy to compute using elementary row operations but hard to evaluate using cofactor expansion

Would a matrix with two rows equal but with values samples from U(0,1), and all other values sampled independently from U(0,1) qualify?

CB
 
what is U(0,1) ?
Please explain me, i couldn't understand.
 
Swati said:
what is U(0,1) ?
Please explain me, i couldn't understand.

U(0,1) - the uniform distribution on [0,1). The suggestion is to use a random matrix generated in the specified way. In one case it would be diagonal, so the co-factor method would give the determinant as the product of the diagonal elements, in the second case with two equal rows row operations would deduce it has zero determinant.

CB
 

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