What is the determinant of this 5x5 matrix and how can it be calculated?

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SUMMARY

The discussion centers on calculating the determinant of a 5x5 matrix using Laplace's expansion and matrix reduction techniques. The matrix provided is:

\[\begin{bmatrix} 1 & 2 & 3 & 3 & 5 \\ 3 & 2 & 1 & 2 & 2 \\ 1 & 2 & 3 & 4 & 5 \\ -1 & 0 & -8 & 1 & 2 \\ 7 & 2 & 1 & 3 & 2 \end{bmatrix}\]

Initially, the user calculated the determinant incorrectly as -379, while the correct value is -224. The discussion highlights the importance of reducing the matrix to smaller dimensions, such as 3x3 or 2x2, for easier calculations.

PREREQUISITES
  • Understanding of determinant calculation methods, specifically Laplace's expansion.
  • Familiarity with matrix reduction techniques to simplify determinant calculations.
  • Basic knowledge of linear algebra concepts, including matrix operations.
  • Ability to interpret and manipulate mathematical notation and equations.
NEXT STEPS
  • Study the process of matrix reduction to 3x3 and 2x2 matrices for determinant calculations.
  • Learn advanced techniques for calculating determinants, such as the Sarrus rule for 3x3 matrices.
  • Practice solving determinant problems using various methods, including cofactor expansion.
  • Explore software tools like MATLAB or Python's NumPy library for automated determinant calculations.
USEFUL FOR

Students in linear algebra courses, educators teaching matrix operations, and anyone seeking to improve their skills in calculating determinants of matrices.

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Homework Statement


so my problem is to calculate the determinant of this matrix
\left[\begin{array}{ccccc}<br /> 1 &amp; 2 &amp; 3 &amp; 3 &amp; 5 \\<br /> 3 &amp; 2 &amp; 1 &amp; 2 &amp; 2 \\<br /> 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\<br /> -1 &amp; 0 &amp; -8 &amp; 1 &amp; 2 \\<br /> 7 &amp; 2 &amp; 1 &amp; 3 &amp; 2<br /> \end{array}\right]



Homework Equations





The Attempt at a Solution


my calculation -> http://img29.imageshack.us/img29/1120/21102009124.th.jpg
i know to use Laplace, but my teacher said me "at first you should reduce matrix to 3x3 or even 2x2 - it's easier to calculate the determinant"
but i don't understand this methode at all.
i've done this problem and my solution is -379 and i know it's wrong (correct is -224)
please help.
 
Last edited by a moderator:
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The third line of your second determinant is incorrect.
 
Your last determinant should be

\left|\begin{array}{cc} -18 &amp; 1\\ -50 &amp;9\end{array}\right|

not

\left|\begin{array}{cc} -18 &amp; 0\\ -50 &amp;9\end{array}\right|
 
yeah, my bad
but thanks now it's correct
thanks a lot :)
 

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