Show that the characteristic equation of a 2 × 2 matrix A can be expressed as 1² – tr(A)A + det(A) = 0, where tr(A) is the trace of A( sum of the elements on the main diagonal)
Show that the characteristic equation of a 2 × 2 matrix A can be expressed as 1² – tr(A)A + det(A) = 0, where tr(A) is the trace of A( sum of the elements on the main diagonal)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show that the characteristic equation of a 2 x 2 matrix A can be expressed as (see image)
![**Expression of the Characteristic Equation for a 2x2 Matrix**
In the study of linear algebra, the characteristic equation of a \(2 \times 2\) matrix \(A\) can be derived and expressed as follows:
\[
\lambda^2 - \text{tr}(A)\lambda + \text{det}(A) = 0,
\]
where:
- \(\lambda\) represents the eigenvalues of the matrix \(A\),
- \(\text{tr}(A)\) denotes the trace of the matrix \(A\), which is the sum of the elements along the main diagonal of \(A\),
- \(\text{det}(A)\) denotes the determinant of the matrix \(A\).
**Additional Explanation:**
The trace of matrix \(A\) (denoted as \(\text{tr}(A)\)) is calculated by summing the values found on the main diagonal of the matrix \(A\):
\[
\text{tr}(A) = a_{11} + a_{22}
\]
where \(a_{11}, a_{22}\) are elements on the main diagonal of \(A\).
The determinant of matrix \(A\) (denoted as \(\text{det}(A)\)) is calculated as:
\[
\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}
\]
where \(a_{11}, a_{12}, a_{21}, a_{22}\) are the elements of the matrix \(A\).
By substituting these values into the characteristic polynomial, we can identify the eigenvalues \(\lambda\) that satisfy the original equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc17e2a4-76f7-46f2-97e3-7d367a16a370%2F750775bf-87ba-4f7b-8f14-f8eef1ef2222%2Fytq2t68n.png&w=3840&q=75)
Transcribed Image Text:**Expression of the Characteristic Equation for a 2x2 Matrix**
In the study of linear algebra, the characteristic equation of a \(2 \times 2\) matrix \(A\) can be derived and expressed as follows:
\[
\lambda^2 - \text{tr}(A)\lambda + \text{det}(A) = 0,
\]
where:
- \(\lambda\) represents the eigenvalues of the matrix \(A\),
- \(\text{tr}(A)\) denotes the trace of the matrix \(A\), which is the sum of the elements along the main diagonal of \(A\),
- \(\text{det}(A)\) denotes the determinant of the matrix \(A\).
**Additional Explanation:**
The trace of matrix \(A\) (denoted as \(\text{tr}(A)\)) is calculated by summing the values found on the main diagonal of the matrix \(A\):
\[
\text{tr}(A) = a_{11} + a_{22}
\]
where \(a_{11}, a_{22}\) are elements on the main diagonal of \(A\).
The determinant of matrix \(A\) (denoted as \(\text{det}(A)\)) is calculated as:
\[
\text{det}(A) = a_{11}a_{22} - a_{12}a_{21}
\]
where \(a_{11}, a_{12}, a_{21}, a_{22}\) are the elements of the matrix \(A\).
By substituting these values into the characteristic polynomial, we can identify the eigenvalues \(\lambda\) that satisfy the original equation.
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