What is the solution for a matrix with a parameter in its determinant?

Click For Summary
SUMMARY

The discussion centers on finding the eigenvalues of the matrix M = [21, 7; 14, 7] by solving the determinant equation |(M - xI)| = 0. The user, Mr. Lard, derived the characteristic polynomial x² - 28x + 49 = 0 using the quadratic formula, resulting in eigenvalues of 14 ± 7√3. The solution method is confirmed as correct, with the term "parameters" clarified as eigenvalues in linear algebra terminology.

PREREQUISITES
  • Understanding of matrix operations, specifically determinants
  • Familiarity with the concept of eigenvalues and eigenvectors
  • Knowledge of the quadratic formula and its application
  • Basic understanding of identity matrices
NEXT STEPS
  • Study the derivation of eigenvalues for 2x2 matrices
  • Learn about the implications of eigenvalues in linear transformations
  • Explore the relationship between eigenvalues and matrix diagonalization
  • Investigate applications of eigenvalues in systems of differential equations
USEFUL FOR

Students studying linear algebra, mathematicians interested in matrix theory, and educators teaching concepts related to eigenvalues and determinants.

lardycake
Messages
1
Reaction score
0

Homework Statement


Hello all,

I have a parameter x of a matrix M = [21, 7
14, 7]
the determinant of
|(M - xI)| = 0

Homework Equations



I is known as the identity matrix

The Attempt at a Solution


My solution seems to be a bit far fetched:

(21 - x)(7-x) - 98 = 0

x2 - 28x + 147 - 98 = 0

x2 - 28x + 49 = 0

using the quadratic formula x = (28 +- root(28x28 - 4(1)(49))) / 2
which I got to be 14 +- 7root(3)

Is this the correct method and answer?

Regards,
Mr Lard
 
Physics news on Phys.org
Yes. And what you are calling parameters are usually called eigenvalues of the matrix.
 

Similar threads

What is the general solution for a DE involving cosh and sinh?
  • · Replies 3 ·
Replies
3
Views
2K
Line parallel to plane with real parameters
  • · Replies 13 ·
Replies
13
Views
1K
ODE using variation of parameters
  • · Replies 6 ·
Replies
6
Views
3K
Is My Matrix Approach to Solving Linear Equations with Parameters Correct?
  • · Replies 3 ·
Replies
3
Views
1K
QR factorization for a 4x4 tridiagonal symmetric matrix
  • · Replies 4 ·
Replies
4
Views
2K
How Does Total Differential Determine Accuracy in Parameter Settings?
  • · Replies 7 ·
Replies
7
Views
2K
Find elements of a matrix such that its determinant is zero
  • · Replies 39 ·
2
Replies
39
Views
6K
How do I get the solution with the matrix exponential method
  • · Replies 11 ·
Replies
11
Views
3K
What Is the Transition Matrix for T in This Transformation?
  • · Replies 4 ·
Replies
4
Views
2K
LinAlg: Determine the value(s) of h such that the matrix....
  • · Replies 8 ·
Replies
8
Views
4K