https://r2.erweima.ai/mathbot/image/1726667728768ly3v66yq.jpg

To solve for the general expression of \( A^n \) for each matrix \( A \), we will analyze each matrix individually.

### Case (a)
Matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \) is a permutation matrix, which swaps the second and third rows.

Let's calculate the first few powers of \( A \) to find a pattern:

- \( A^1 = A \)
- \( A^2 = A \cdot A \)
- \( A^3 = A \cdot A^2 \)

Since this matrix is a permutation matrix, we should observe a cyclical pattern.

### Case (b)
Matrix \( A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \) is neither diagonal nor a typical permutation matrix.

Here, we can try to diagonalize \( A \), or use Jordan canonical form if diagonalization is not possible. Let's compute the characteristic polynomial, eigenvalues, and eigenvectors, and then we can generalize the powers of \( A \).

### Case (c)
Matrix \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \) is a block triangular matrix, with eigenvalue \( 1 \) of multiplicity 3.

For this matrix, we can also attempt to diagonalize it or express it in Jordan canonical form, as it has a single eigenvalue. We will check how the generalized powers behave.

---

Now, I'll compute the general expressions for these matrices. Let me proceed with calculations.Here are the results for the first few powers of the matrices:

### Case (a)
Matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \):

- \( A^1 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \)
- \( A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \) (identity matrix)
- \( A^3 = A \) again, indicating a cycle every 2 powers.

So, the general expression for \( A^n \) is:
- \( A^n = A \) if \( n \) is odd,
- \( A^n = I \) (the identity matrix) if \( n \) is even.

### Case (b)
Matrix \( A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \):

- \( A^1 = A \)
- \( A^2 = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
- \( A^3 = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} \)
- \( A^4 = \begin{pmatrix} 2 & 3 & 0 \\ 3 & 5 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)

This pattern suggests that for higher powers, the matrix elements grow according to Fibonacci-like numbers in the upper-left 2x2 block, while the bottom right corner alternates between 1 and -1. We can generalize the growth of the elements but would need to explore further to formalize a general closed form.

### Case (c)
Matrix \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \):

- \( A^1 = A \)
- \( A^2 = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
- \( A^3 = \begin{pmatrix} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
- \( A^4 = \begin{pmatrix} 1 & 4 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)

The general expression for \( A^n \) is:
\[ A^n = \begin{pmatrix} 1 & n & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
This follows from the observation that the \( (1,2) \)-entry increases linearly with \( n \), while all other entries remain constant.

---

Would you like further details or explanations on any specific case?

Here are some follow-up questions to deepen your understanding:
1. How do you calculate the inverse of matrix \( A \) in case (a)?
2. What is the characteristic polynomial of matrix \( A \) in case (b)?
3. How would you prove the general expression for \( A^n \) in case (c)?
4. Can matrix \( A \) in case (b) be diagonalized, or does it need a Jordan form?
5. How do powers of block matrices, such as in case (b), behave in general?

**Tip:** Understanding the cyclic behavior of permutation matrices, like in case (a), is essential in simplifying matrix exponentiation problems.

Halla la expresión general de A^n en los siguientes casos: (a) A = [[1, 0, 0], [0, 0, 1], [0, 1, 0]], (b) A = [[0, 1, 0], [1, 1, 0], [0, 0, -1]], (c) A = [[1, 1, 0], [0, 1, 0], [0, 0, 1]].

Solve for the general expression of A^n for three different matrices: a permutation matrix, a matrix with Jordan form characteristics, and a block triangular matrix. Detailed solution using matrix exponentiation, eigenvalues, and diagonalization concepts. Suitable for linear algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation