Determinant - Proof for distinct real numbers

In summary, the determinant of a matrix is a numerical value that can be calculated using various methods, such as the cofactor expansion method, the diagonal method, or the Gaussian elimination method. It can be positive, negative, or zero depending on the arrangement of the elements in the matrix. When proving the determinant, it is important for the real numbers to be distinct to accurately represent the properties of the matrix. The determinant has many applications in fields such as engineering, physics, economics, and statistics. It is used to solve systems of equations, calculate areas and volumes, determine matrix invertibility, and analyze data sets for linear dependence.
  • #1
Dethrone
717
0
View attachment 4292

I was able to prove a), but I am unsure how to prove b. Is there some sort of geometric interpretation I should be aware of?
 

Attachments

  • determinant 3.PNG
    determinant 3.PNG
    18.7 KB · Views: 55
Physics news on Phys.org
  • #3
Rido12 said:
I was able to prove a), but I am unsure how to prove b. Is there some sort of geometric interpretation I should be aware of?

Hey Rido! ;)

One way is apply the property of a determinant that it is 0 iff its columns form a linear combination. (Thinking)
 

1. What is the determinant of a matrix?

The determinant of a matrix is a numerical value that can be calculated from the elements of the matrix. It is used to determine various properties of the matrix, such as whether it is invertible or singular.

2. How is the determinant calculated?

The determinant of a matrix can be calculated using various methods, such as the cofactor expansion method, the diagonal method, or the Gaussian elimination method. Each method involves a specific set of mathematical operations on the elements of the matrix.

3. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the arrangement of the elements in the matrix and can be positive, negative, or zero.

4. Why is it important for the real numbers in a matrix to be distinct when proving the determinant?

When proving the determinant of a matrix, it is important for the real numbers to be distinct because if there are repeated numbers, the determinant will be equal to zero. This can make the proof more complicated and may not accurately represent the properties of the matrix.

5. What are some applications of the determinant in real life?

The determinant has various applications in fields such as engineering, physics, economics, and statistics. It is used to solve systems of equations, calculate areas and volumes, determine whether a matrix is invertible, and analyze data sets for linear dependence.

Similar threads

Replies
4
Views
603
  • Linear and Abstract Algebra
Replies
15
Views
4K
  • Linear and Abstract Algebra
Replies
1
Views
920
  • Linear and Abstract Algebra
Replies
17
Views
1K
  • Linear and Abstract Algebra
Replies
17
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
350
  • Linear and Abstract Algebra
Replies
13
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
960
  • Linear and Abstract Algebra
Replies
11
Views
1K
  • Linear and Abstract Algebra
Replies
26
Views
6K
Back
Top