You have the determinant of the matrix $A - \lambda I$ given as a polynomial expression. Let's break it down step by step to better understand it and simplify the equation.

The expression is:

$\det(A - \lambda I) = (5 - \lambda) \left( (-2 - \lambda)(5 - \lambda) - 3(-4) \right) + 3 \left( 4(5 - \lambda) - 3(4) \right) + (-3) \left( 4(-4) - (-2 - \lambda)(4) \right)$

This looks like it's from a 3x3 matrix, where you are expanding the determinant using cofactor expansion along the first row.

Let's simplify each part of the expression one by one.

1. First term: $(5 - \lambda) \left( (-2 - \lambda)(5 - \lambda) - 3(-4) \right)$

• $(-2 - \lambda)(5 - \lambda) = -2(5 - \lambda) - \lambda(5 - \lambda) = -10 + 2\lambda - 5\lambda + \lambda^2 = \lambda^2 - 3\lambda - 10$
• $3(-4) = -12$
• So, the inside of the parentheses becomes: $(\lambda^2 - 3\lambda - 10) - (-12) = \lambda^2 - 3\lambda + 2$
• Now, multiply by $(5 - \lambda)$: $(5 - \lambda)(\lambda^2 - 3\lambda + 2) = 5(\lambda^2 - 3\lambda + 2) - \lambda(\lambda^2 - 3\lambda + 2)$ $= 5\lambda^2 - 15\lambda + 10 - \lambda^3 + 3\lambda^2 - 2\lambda$ $= -\lambda^3 + 8\lambda^2 - 17\lambda + 10$

2. Second term: $3 \left( 4(5 - \lambda) - 3(4) \right)$

• $4(5 - \lambda) = 20 - 4\lambda$
• $3(4) = 12$
• So, inside the parentheses becomes: $(20 - 4\lambda) - 12 = 8 - 4\lambda$
• Now, multiply by $3$: $3(8 - 4\lambda) = 24 - 12\lambda$

3. Third term: $(-3) \left( 4(-4) - (-2 - \lambda)(4) \right)$

• $4(-4) = -16$
• $(-2 - \lambda)(4) = -8 - 4\lambda$
• So, inside the parentheses becomes: $(-16) - (-8 - 4\lambda) = -16 + 8 + 4\lambda = -8 + 4\lambda$
• Now, multiply by $-3$: $-3(-8 + 4\lambda) = 24 - 12\lambda$

Combine all terms:

Now let's put everything together:

$\det(A - \lambda I) = -\lambda^3 + 8\lambda^2 - 17\lambda + 10 + 24 - 12\lambda + 24 - 12\lambda$ $= -\lambda^3 + 8\lambda^2 - 41\lambda + 58$

Thus, the determinant of $A - \lambda I$ is:

$\boxed{-\lambda^3 + 8\lambda^2 - 41\lambda + 58}$

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Would you like a more detailed explanation of any step? Or perhaps help with finding the eigenvalues or solving this determinant for $\lambda$?

Further Questions:

1. How do we use this characteristic polynomial to find the eigenvalues of matrix $A$?
2. What is the relation between the determinant and the eigenvalues of a matrix?
3. Can you show the steps to solve a cubic equation like this for $\lambda$?
4. How would you handle higher-degree determinants using similar methods?
5. How does the cofactor expansion method apply to other matrices like 4x4 matrices?

Tip:

If you want to check your work, consider using a numerical solver for the cubic equation or a symbolic math tool to confirm the roots of the determinant expression.