已知如下矩阵方程，回答下述问题。

Given the matrix equation below, answer the following questions.

1.哪个是方程中的特征向量？Which is the eigen vector in the equation?

选择
[選擇]
左侧的 2x2 矩阵。The 2x2 matrix on the left.
值为 (1, 1) 的向量 The vector with value (1, 1)
值为 (2, 2) 的向量 The vector with value (2, 2)
值为 (5, 5) 的向量 The vector with value (5, 5)

2.特征值是多少？What is the eigen value?

选择
[選擇]
1
2
3
4
5

Question

已知如下矩阵方程，回答下述问题。

Given the matrix equation below, answer the following questions.

1.哪个是方程中的特征向量？Which is the eigen vector in the equation?

选择 
[選擇]
左侧的 2x2 矩阵。The 2x2 matrix on the left.
值为 (1, 1) 的向量 The vector with value (1, 1)
值为 (2, 2) 的向量 The vector with value (2, 2)
值为 (5, 5) 的向量 The vector with value (5, 5)

2.特征值是多少？What is the eigen value?

选择 
[選擇]
1
2
3
4
5

BlackTom AI · Accepted Answer

An eigenvector v satisfies Av = λv for some scalar λ. Check which given vector makes this true for A = [[2,3],[3,2]].

1) Relevant concepts
- Definition: If A is a square matrix, a nonzero vector v is an eigenvector when Av is a scalar multiple of v; that scalar is the eigenvalue λ.
- Matrix-vector multiplication and scalar multiplication are used to verify the relation Av = λv.

2) Step-by-step reasoning and computation
Compute A times the candidate vector (1, 1)^T:
- First component: 2·1 + 3·1 = 2 + 3 = 5
- Second component: 3·1 + 2·1 = 3 + 2 = 5
So A(1,1)^T = (5,5)^T.

3) Intermediate steps shown as equality to λv
We can write (5,5)^T = 5·(1,1)^T. Therefore A(1,1)^T = 5·(1,1)^T, which matches the eigenvector/eigenvalue form Av = λv with λ = 5.

4) Why the correct answers are correct
Because multiplying the matrix by (1,1) produces a vector that is exactly 5 times (1,1), (1,1) is an eigenvector and 5 is the associated eigenvalue. This directly confirms the equation shown in the problem.

5) Why the other choices are incorrect (brief)
- The 2x2 matrix on the left is not a vector; it is the linear operator A, not an eigenvector.
- The vectors (2,2) and (5,5) are scalar multiples of (1,1) and therefore are also eigenvectors in general; however, the specific vector appearing in the equation is (1,1). The equation explicitly uses (1,1) on the right-hand side of the multiplication; (2,2) and (5,5) are not the vector shown in the original equation.
- The eigenvalue options 1–4 are not equal to the scalar factor found (5), so they are incorrect.

Conclusion
The eigenvector shown in the equation is (1,1) and the corresponding eigenvalue is 5, matching the provided answers.

Similar Questions

Question textSuppose a [math: 4×4] matrix has four eigenvalues [math: −1], [math: 2], [math: 1+i] and [math: 1−i]. What is the maximum number of linearly independent eigenvectors corresponding to the eigenvalue [math: 2]?Answer: Answer 1 Question 2[input]

Question textSuppose a [math: 4×4] matrix has four eigenvalues [math: −1], [math: 2], [math: 1+i] and [math: 1−i]. What is the maximum number of linearly independent eigenvectors corresponding to the eigenvalue [math: 2]?Answer: Answer 1 Question 2[input]

特征向量的特征值 𝜆 是指 The eigenvalue 𝜆 of an eigen vector is

Let 𝐴 = [ 𝑎 𝑏 𝑐 𝑑 ] , where 𝑎 , 𝑏 , 𝑐 , and 𝑑 are real numbers. If [ 1 1 ] is an eigenvector of 𝐴 corresponding to the eigenvalue 𝜆 = 4 , find the value of 𝑎 + 𝑏 + 𝑐 + 𝑑 .

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