How Do Eigenvalues Determine the Value of a Matrix's Determinant?

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  • #1
deana
3
0

Homework Statement

Let A be the matrix with eigenvalues x1 = 2, x2 = 1, x3 = 1/2 , x4 = 10

and corresponding eigenvectors v1: <1,-1,1,0>, v2: <1,-1,0,0>, v3: <1,0,0,1>, v4: <0,0,1,1>

Calculate |A|

Homework Equations



See above

The Attempt at a Solution



I'm not really sure how to start this problem but i know that:
For nxn matrices X, Y , Z
|XYZ| = |X| |Y| |Z| and |X^ (-1)|= 1 / |X|
Maybe I could use this to solve the problem?

Any input or suggestions about how to start this problem would be helpful!
Thanks!:)
 
Last edited:
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  • #2
What does the matrix of your linear transformation look like if you express it in the basis {v1,v2,v3,v4}?
 
  • #3
Do you know the relationship between the eigenvalues of a matrix and the determinant of that matrix? It is a standard result. If it is not in your textbook or course notes, it can certainly be found through Google.

RGV
 

FAQ: How Do Eigenvalues Determine the Value of a Matrix's Determinant?

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 represent certain properties of the matrix, such as whether the matrix is invertible or singular.

2. How is the determinant of a matrix calculated?

The determinant of a matrix is calculated using a specific formula depending on the size of the matrix. For a 2x2 matrix, the formula is ad-bc, where a, b, c, and d are the elements of the matrix. For larger matrices, the calculation involves expanding along a row or column and performing calculations on smaller submatrices.

3. What is the significance of the determinant in linear algebra?

The determinant is an important concept in linear algebra as it is used to determine various properties of a matrix, such as whether it is invertible or singular. It is also used in solving systems of linear equations, finding eigenvalues and eigenvectors, and in various other applications in mathematics and science.

4. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative, positive, or zero. The sign of the determinant is determined by the arrangement of the elements in the matrix and does not affect its value or calculations involving the determinant.

5. How is the determinant used in solving systems of linear equations?

The determinant is used in solving systems of linear equations by determining whether the system has a unique solution, no solution, or infinitely many solutions. If the determinant of the coefficient matrix is non-zero, then the system has a unique solution. If the determinant is zero, then the system has either no solution or infinitely many solutions, which can be further determined by solving the system.

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