Linear Algebra and Determinant

[Swati]

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

 

[CaptainBlack]

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

 

[Swati]

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

 

[CaptainBlack]

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

 

[blue_raver22]

Linear algebra is a fundamental branch of mathematics that deals with the study of vector spaces and linear transformations. It plays a crucial role in many fields, including physics, engineering, and computer science.

Determinants are an important concept in linear algebra, used to determine the invertibility and the solutions of linear systems of equations. They are computed using various methods, including cofactor expansion and elementary row operations.

In response to the given content, I will provide two matrices that demonstrate the difference in evaluating determinants using cofactor expansion and elementary row operations.

(a) The matrix [1 0 0 0; 0 2 0 0; 0 0 3 0; 0 0 0 4] has a determinant of 24, which is easy to compute using cofactor expansion. The determinant is simply the product of the diagonal elements, as all other elements are zero. However, evaluating this determinant using elementary row operations would require multiple steps and calculations, making it hard to evaluate.

(b) The matrix [1 2 3 4; 2 4 6 8; 3 6 9 12; 4 8 12 16] has a determinant of 0, which is easy to compute using elementary row operations. By performing row operations, we can reduce this matrix to a row of zeros, making the determinant equal to 0. However, using cofactor expansion to evaluate this determinant would require calculating the determinants of multiple submatrices, making it a more complicated and time-consuming process.

In conclusion, the choice of method for evaluating determinants depends on the given matrix. Some matrices may have determinants that are easy to compute using cofactor expansion, while others may require elementary row operations. As scientists, it is important to understand and utilize both methods to effectively solve problems in linear algebra.