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
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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.
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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