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The problem asks us to evaluate \( f(A) \), where the matrix \( A \) and the function \( f(x) \) are given as:

\[
A = \begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}
\]

\[
f(x) = x^{100} + 5x^{99} + 6x + 3
\]

To solve \( f(A) \), we need to compute the matrix function by substituting the matrix \( A \) into the polynomial \( f(x) \).

### Step 1: Compute the powers of the matrix \( A \)
We need to find \( A^{100} \), \( A^{99} \), and \( A \).

1. **First, calculate \( A^2 \):**

\[
A^2 = A \times A = \begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix} \times \begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix} 
\]

\[
A^2 = \begin{pmatrix} (2)(2) + (-1)(3) & (2)(-1) + (-1)(-2) \\ (3)(2) + (-2)(3) & (3)(-1) + (-2)(-2) \end{pmatrix} = \begin{pmatrix} 4 - 3 & -2 + 2 \\ 6 - 6 & -3 + 4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

Notice that \( A^2 = I \), the identity matrix. Therefore:

\[
A^3 = A \times A^2 = A \times I = A
\]

This indicates that for any power \( n \), \( A^n = A \) if \( n \) is odd, and \( A^n = I \) if \( n \) is even.

### Step 2: Simplify the function using the above result
Since \( A^2 = I \), the higher powers simplify as follows:

- \( A^{100} = (A^2)^{50} = I \)
- \( A^{99} = A^{98} \times A = I \times A = A \)

Thus:

\[
f(A) = I + 5A + 6A + 3I
\]

Simplifying further:

\[
f(A) = 4I + 11A
\]

### Step 3: Substitute the values of \( A \) and \( I \):

\[
f(A) = 4\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + 11\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}
\]

\[
f(A) = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} + \begin{pmatrix} 22 & -11 \\ 33 & -22 \end{pmatrix} = \begin{pmatrix} 26 & -11 \\ 33 & -18 \end{pmatrix}
\]

Thus, the value of \( f(A) \) is:

\[
f(A) = \begin{pmatrix} 26 & -11 \\ 33 & -18 \end{pmatrix}
\]

Would you like more details or have any other questions?

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**Relative questions:**

1. How do you find the characteristic polynomial of matrix \( A \)?
2. What are the eigenvalues of matrix \( A \)?
3. Can we diagonalize the matrix \( A \), and if so, how?
4. What is the Cayley-Hamilton theorem, and how is it applicable here?
5. How does the identity matrix \( I \) affect matrix operations?

**Tip:** When working with matrix polynomials, always check for simplifications like powers of \( A \) reducing to \( I \) or \( A \). This can greatly simplify the calculations.

Cho ma trận A = \begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}, biết f(x) = x^{100} + 5x^{99} + 6x + 3. Tính f(A)

Explore the step-by-step solution to evaluating the matrix function f(A), where A is a 2x2 matrix and f(x) is a high-degree polynomial. This problem involves matrix algebra, polynomial functions, and properties of matrix powers. Suitable for university-level students.

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