1−λ
1
−4
​

1
−3−λ
−4
​

3
−2
−4−λ
​

It looks like you're dealing with a matrix equation or a characteristic equation involving \(\lambda\). Let's start by analyzing the matrix. Based on how you've structured the entries, it seems like you're working with a \(3 \times 3\) matrix:

\[
\begin{pmatrix}
1 - \lambda & 1 & -4 \\
1 & -3 - \lambda & -4 \\
3 & -2 & -4 - \lambda
\end{pmatrix}
\]

### Determinant of the Matrix
To find the characteristic equation, which is typically what you're looking for with \(\lambda\) in the matrix, you would set the determinant of this matrix equal to zero:

\[
\text{det}(A - \lambda I) = 0
\]

Here, \(A\) is the matrix and \(I\) is the identity matrix.

### Calculation of the Determinant
Let's compute the determinant of the matrix:

\[
\text{det} = 
\begin{vmatrix}
1 - \lambda & 1 & -4 \\
1 & -3 - \lambda & -4 \\
3 & -2 & -4 - \lambda
\end{vmatrix}
\]

This determinant can be calculated using cofactor expansion. Let's go through the steps:

1. **Expand along the first row:**
   \[
   \text{det} = (1 - \lambda) \begin{vmatrix} -3 - \lambda & -4 \\ -2 & -4 - \lambda \end{vmatrix}
   - 1 \cdot \begin{vmatrix} 1 & -4 \\ 3 & -4 - \lambda \end{vmatrix}
   + (-4) \cdot \begin{vmatrix} 1 & -3 - \lambda \\ 3 & -2 \end{vmatrix}
   \]

2. **Calculate each of the \(2 \times 2\) determinants:**

   - For the first determinant:
     \[
     \begin{vmatrix} -3 - \lambda & -4 \\ -2 & -4 - \lambda \end{vmatrix}
     = (-3 - \lambda)(-4 - \lambda) - (-4)(-2)
     = \lambda^2 + 7\lambda + 4
     \]
     
   - For the second determinant:
     \[
     \begin{vmatrix} 1 & -4 \\ 3 & -4 - \lambda \end{vmatrix}
     = 1(-4 - \lambda) - (-4)(3)
     = -\lambda - 4 + 12
     = 8 - \lambda
     \]

   - For the third determinant:
     \[
     \begin{vmatrix} 1 & -3 - \lambda \\ 3 & -2 \end{vmatrix}
     = 1(-2) - (-3 - \lambda)(3)
     = -2 + 9 + 3\lambda
     = 7 + 3\lambda
     \]

3. **Substitute back into the determinant expansion:**
   \[
   \text{det} = (1 - \lambda)(\lambda^2 + 7\lambda + 4) - (8 - \lambda) - 4(7 + 3\lambda)
   \]

4. **Expand and simplify:**
   \[
   \text{det} = (1 - \lambda)(\lambda^2 + 7\lambda + 4) - (8 - \lambda) - (28 + 12\lambda)
   \]
   Simplifying further, we get the characteristic polynomial.

Would you like to proceed with the expansion, or do you have any specific questions at this point?

### Related Questions
1. How do we determine eigenvalues from the characteristic equation?
2. What is the significance of eigenvalues in linear algebra?
3. How do we find eigenvectors corresponding to a given eigenvalue?
4. What is the role of determinants in solving systems of linear equations?
5. How does the characteristic equation relate to matrix diagonalization?

### Tip
Always double-check the signs and terms when expanding determinants, especially in cofactor expansions, as small errors can lead to incorrect solutions.

Explore the detailed analysis of a matrix equation involving λ. Learn how to compute the determinant and derive the characteristic polynomial. This problem covers matrix algebra, determinants, and characteristic equations.

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