Question text 4Marks If [math: [31a]]\left[\matrix{ 3 \cr 1 \cr a } \right] is an eigenvector of [math: A=[3−11−130151],] A = \left[\matrix{ 3 & -1 & 1 \cr -1 & 3 & 0 \cr 1 & 5 & 1 } \right], then [math: a=] Answer 1[input], and the corresponding eigenvalue is [math: λ=]\lambda= Answer 2[input].Notes Report question issue Question 4 Notes多项填空题
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Question

Question text 4Marks If [math: [31a]]\left[\matrix{ 3 \cr 1 \cr a } \right] is an eigenvector of [math: A=[3−11−130151],] A = \left[\matrix{ 3 & -1 & 1 \cr -1 & 3 & 0 \cr 1 & 5 & 1 } \right], then [math: a=] Answer 1[input], and the corresponding eigenvalue is [math: λ=]\lambda= Answer 2[input].Notes Report question issue Question 4 Notes多项填空题

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BlackTom AI · Accepted Answer

We are given the matrix A and a vector form [3, 1, a]^T claimed to be an eigenvector, with eigenvalue λ, so Av = λv must hold.

First, write the matrix A and multiply by v = [3, 1, a]^T:
- Av first component: 3*3 + (-1)*1 + 1*a = 9 - 1 + a = 8 + a
- Av second component: (-1)*3 + 3*1 + 0*a = -3 + 3 + 0 = 0
- Av third component: 1*3 + 5*1 + 1*a = 3 + 5 + a = 8 + a

Since Av = λv = [3λ, λ, aλ]^T, we equate corresponding components:
- From the second component: 0 = λ. This immediately gives λ = 0.
- With λ = 0, the first component gives: 8 + a = 3λ = 0, so a = -8.
- The third component gives: 8 + a = aλ = a*0 = 0, which again yields a = -8, consistent with the previous equation.

Thus, the vector [3, 1, a]^T is an eigenvector corresponding to the eigenvalue λ = 0 when a = -8.

Summary of the reasoning steps:
- Compute Av components to set up equations for λ and a.
- Use the fact that the second component forces λ = 0.
- Substitute λ = 0 into the first (and third) components to solve for a.
- Check consistency of a from all components; it should be -8 in this case.

Common potential missteps to avoid:
- Assuming a nonzero λ from the second component; the second component directly constrains λ via 0 = λ.
- Forgetting to also enforce the third component, which must be consistent with aλ as well.
- Not checking consistency between all components; here both first and third components collapse to the same requirement a = -8 once λ = 0 is known.