#6Please answer in detail ### Eigenvalues of a General 2x2 Matrix#### Matrix DefinitionWe 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 Description1. **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).#### Explanation1. **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: \[ (

Name: #6Please answer in detail ### Eigenvalues of a General 2x2 Matrix####
Uploaded: 2024-03-05T11:52:21.000Z
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Description: #6Please answer in detail ### Eigenvalues of a General 2x2 Matrix#### Matrix DefinitionWe 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