How Do You Calculate the Determinant of This Complex Matrix?

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SUMMARY

The discussion focuses on calculating the determinant of a complex matrix structured with specific row patterns. The matrix in question features a sequence of integers in the first row, followed by increasingly negative integers in subsequent rows, culminating in a row of zeros. Key techniques for calculating determinants include the ability to add multiples of rows or columns without altering the determinant's value. Examples provided illustrate the calculation of determinants for smaller matrices, establishing foundational methods applicable to the larger matrix.

PREREQUISITES
  • Understanding of matrix notation and structure
  • Familiarity with determinant properties and rules
  • Basic knowledge of linear algebra concepts
  • Experience with matrix operations, including row and column manipulation
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn techniques for row reduction and their impact on determinants
  • Explore the cofactor expansion method for calculating determinants
  • Practice calculating determinants of larger matrices using software tools like MATLAB or Python's NumPy
USEFUL FOR

Students studying linear algebra, mathematics educators, and anyone seeking to improve their skills in calculating determinants of complex matrices.

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Homework Statement


I have to find a determinant for
1 2 3 ... n
-1 0 3 ... n
-1 -2 0 ... n
...
-1 -2 -3 ... 0
but I have very little clue how to proceed, because the mathematics material that I was given is very vague about this. Any help would be greatly appreciated.


Homework Equations





The Attempt at a Solution

 
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Does -1 -2 -3 ... 0 represent the final row of the matrix meaning the the lowest right member is 0?
 
Welcome to PF, Helloes! :smile:

The main rule about calculating determinants, is that you can add a multiple of a row to another row, without changing the determinant.
Same thing for columns: you can add (or subtract) a multiple of one column to another column.

Now let's try a couple of determinants.

What is |1|?

What is [itex]\begin{vmatrix}1&2\\-1&0\end{vmatrix}[/itex]?

What is [itex]\begin{vmatrix}1&2&3\\-1&0&3\\-1&-2&0\end{vmatrix}[/itex]?

What is [itex]\begin{vmatrix}1&2&3&4\\-1&0&3&4\\-1&-2&0&4\\-1&-2&-3&0\end{vmatrix}[/itex]?
 

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