6) We call the matrix below the general 2 × 2 matrix because its entries could be any real numbers. a Find a formula (which will obviously involve a, b, c, and d) for the eigenvalues of the general 2×2 matrix. Find an equation which describes those 2x2 matrices that have only one eigenvalue, and gives three examples of such matrices (only one of your examples may be triangular)
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![### Eigenvalues of a General 2x2 Matrix
#### Matrix Definition
We call the matrix below the **general 2 × 2 matrix** because its entries could be any real numbers.
\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
#### Problem Description
1. **Formula for Eigenvalues**:
- Find a formula (which will obviously involve \(a, b, c,\) and \(d\)) for the eigenvalues of the general 2 × 2 matrix.
2. **Equation for Matrices with Only One Eigenvalue**:
- Find an equation that describes those 2 × 2 matrices that have only one eigenvalue.
3. **Examples**:
- Provide three examples of such matrices (only one of your examples may be triangular).
#### Explanation
1. **Eigenvalue Formula**:
To find the eigenvalues of a 2 × 2 matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\), we use the characteristic equation:
\[
\text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0
\]
where \(\mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) and \(\mathbf{I}\) is the identity matrix \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\).
The characteristic equation becomes:
\[
\left| \begin{array}{cc} a - \lambda & b \\ c & d - \lambda \end{array} \right| = 0
\]
Simplifying, we get the quadratic equation:
\[
\lambda^2 - (a + d)\lambda + (ad - bc) = 0
\]
The eigenvalues \(\lambda\) are the solutions to this quadratic equation:
\[
\lambda = \frac{(a + d) \pm \sqrt{(a + d)^2 - 4(ad - bc)}}{2}
\]
2. **Equation for Matrices with One Eigenvalue**:
For a matrix to have only one eigenvalue, the discriminant of the quadratic equation must be zero:
\[
(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecab317-fab7-42bc-8c65-e82787a88e59%2F16476ce0-f00e-439f-8bd7-a3e7ad5c962d%2Fuiyjebf_processed.png&w=3840&q=75)
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