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Problem 222

# [A] Find the minimum polynomial $$\mathrm{m}(\lambda)$$ of the matrix $\mathrm{A}=\begin{array}{cccc}2 & 1 & 0 & 0 \\ & 0 & 2 & 0 & 0 \mid \\ & 0 & 0 & 2 & 0 \\ & 0 & 0 & 0 & 5\end{array}$ [B] Let $$\mathrm{A}$$ be a 3 by 3 matrix over the real field cannot be a zero of the polynomial, $$\varphi(\lambda)=\lambda^{2}+1$$

Expert verified
The minimum polynomial of the given matrix is $$m(\lambda) = (\lambda - 2)^{2}(\lambda - 5)$$. For a 3x3 matrix A, it is impossible for φ(λ) to be the characteristic polynomial. An example of a 3x3 real matrix that is not a zero of φ(λ) is the identity matrix with the characteristic polynomial (λ - 1)^3.
See the step by step solution

## Step 1: Find Eigenvalues and their Multiplicities

Since the given matrix is diagonal, the eigenvalues are the diagonal entries. So, we have three eigenvalues: λ1 = 2, λ2 = 2, and λ3 = 5. Since λ1 and λ2 are equal, let's combine them as just λ1 with algebraic multiplicity 2.

## Step 2: Construct the Minimum Polynomial

Now, we will construct the minimum polynomial m(λ) using the eigenvalues and their algebraic multiplicities. m(λ) must be the product of unique linear factors corresponding to distinct eigenvalues, each raised to their algebraic multiplicity. m(λ) = (λ - λ1)^2(λ - λ3) m(λ) = (λ - 2)^2(λ - 5) So the minimum polynomial is: $m(\lambda) = (\lambda - 2)^{2}(\lambda - 5)$ [B]

## Step 1: Find the Characteristic Polynomial

A 3x3 matrix A will have a characteristic polynomial of degree 3. We will call it χ(λ) = c_0 + c_1λ + c_2λ^2 + λ^3.

## Step 2: Identify Matrices Where φ(λ) is not the Characteristic Polynomial

We know that φ(λ) = λ^2 + 1. For a 3x3 matrix A, it is impossible for φ(λ) to be the characteristic polynomial. This is because the characteristic polynomial of a 3x3 matrix must have degree 3 while φ(λ) has degree 2.

## Step 3: Give an Example

Consider the 3x3 identity matrix: $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ The characteristic polynomial of A is: χ(λ) = (λ - 1)^3 This matrix A is an example of a 3x3 real matrix that is not a zero of φ(λ).

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