Uncovering the Pattern: Factoring and Vanishing Points in Determinants

  • Thread starter astonmartin
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In summary, there is a pattern for finding determinants of matrices with a specific form. For a 2x2 matrix, the determinant is y - x. For a 3x3 matrix, the determinant is xy^2 - yx^2 - xz^2 +zx^2 + yz^2 - zy^2, which can be factored using the roots. The determinant vanishes at x=y, x=z, and y=x, each of which provides a factor of the determinant.
  • #1
astonmartin
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Homework Statement




Alright so apparently there is some pattern in finding these determinants:

for the 2x2, the determinant of
|1 1|
|x y| is y - x

for 3x3, the determinant of

|1 1 1 |
|x y z |
|x^2, y^2, z^2| is xy^2 - yx^2 - xz^2 +zx^2 + yz^2 - zy^2

Apparently that can be factored (not sure how), and using the roots, there will be a pattern that you can observe for finding a determinant of a 4x4, 5x5, etc of the same form (1, x, x^2, x^3, etc.)

How do you factor the 3x3 determinant, and what is the pattern? Does the determinant vanish at some point?
 
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  • #2
Start with the 3x3 case. The pattern is that the determinant vanishes in the cases x=y, x=z and y=x. Do you see why? Each of those tells you a factor of the determinant.
 

What's the pattern here?

The pattern in a given set of data refers to a recurring sequence or trend that exists within the data points.

How do I identify the pattern?

To identify a pattern in data, you can visually analyze the data points and look for any recurring sequences or trends. You can also use mathematical tools such as graphs, charts, and statistical analysis to help identify patterns.

Why is it important to identify patterns?

Identifying patterns in data can provide valuable insights and help make predictions about future trends or events. It can also aid in problem-solving and decision-making processes.

What are some common types of patterns?

Some common types of patterns include linear, quadratic, exponential, periodic, and random patterns. These patterns can be identified through various mathematical methods and techniques.

Can patterns change over time?

Yes, patterns can change over time as data points are added or removed from a data set. It is important to regularly analyze and update patterns to account for any changes that may occur.

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