https://r2.erweima.ai/mathbot/image/1726536148967uycq5s4a.jpg

The problem asks to find the value of \( x \) such that the matrix \( A = \begin{pmatrix} 5 & 0 \\ 2 & x \end{pmatrix} \) is a root of the matrix equation \( f(X) = X^2 - 25 \), where \( X \) is a matrix.

### Step 1: Interpret the matrix equation.
We are given:
\[
f(X) = X^2 - 25
\]
So, \( A \) is a root of this equation if:
\[
A^2 - 25I = 0
\]
Where \( I \) is the identity matrix of the same size as \( A \), and \( 25I = 25 \cdot I = 25 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 25 & 0 \\ 0 & 25 \end{pmatrix} \).

Thus, we need to compute \( A^2 \) and solve the equation:
\[
A^2 = 25I
\]

### Step 2: Compute \( A^2 \).
The matrix \( A \) is:
\[
A = \begin{pmatrix} 5 & 0 \\ 2 & x \end{pmatrix}
\]
Now, calculate \( A^2 = A \cdot A \):
\[
A^2 = \begin{pmatrix} 5 & 0 \\ 2 & x \end{pmatrix} \cdot \begin{pmatrix} 5 & 0 \\ 2 & x \end{pmatrix}
\]
Performing the matrix multiplication:
\[
A^2 = \begin{pmatrix}
5 \cdot 5 + 0 \cdot 2 & 5 \cdot 0 + 0 \cdot x \\
2 \cdot 5 + x \cdot 2 & 2 \cdot 0 + x \cdot x
\end{pmatrix}
= \begin{pmatrix} 25 & 0 \\ 10 + 2x & x^2 \end{pmatrix}
\]

### Step 3: Set up the equation.
For \( A^2 = 25I \), we need:
\[
\begin{pmatrix} 25 & 0 \\ 10 + 2x & x^2 \end{pmatrix} = \begin{pmatrix} 25 & 0 \\ 0 & 25 \end{pmatrix}
\]
From this, we get two equations:
1. \( 10 + 2x = 0 \)
2. \( x^2 = 25 \)

### Step 4: Solve the equations.
1. From \( 10 + 2x = 0 \), we find \( x = -5 \).
2. From \( x^2 = 25 \), we get two solutions: \( x = 5 \) and \( x = -5 \).

### Step 5: Conclusion.
The only solution that satisfies both equations is \( x = -5 \).

Thus, the value of \( x \) is \( \boxed{-5} \).

Would you like further details or explanations?

### Related Questions:
1. How do you verify if a matrix is a root of a polynomial equation?
2. What is the geometric meaning of matrix roots in relation to transformations?
3. How does matrix multiplication work for non-diagonal matrices?
4. Can a matrix have more than one root for a given polynomial?
5. How can the Cayley-Hamilton theorem be applied to this type of problem?

### Tip:
When solving matrix equations, always check the dimensions and ensure that the operations (such as multiplication) are valid for the matrices involved.

Sea A = (5  0, 2  x); determinar el valor de x para que A sea una raíz de f(X) = X^2 - 25.

Learn how to solve the problem of finding the value of x such that matrix A is a root of the matrix equation f(X) = X^2 - 25. This problem involves matrix multiplication and solving quadratic equations. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation