Question text 4Marks If [math: [31a]] is an eigenvector of [math: A=[3−11−130151],] then [math: a=] Answer 1[input], and the corresponding eigenvalue is [math: λ=] Answer 2[input].Notes Report question issue Question 3 Notes

Question

BlackTom AI · Accepted Answer

Eigenvector condition: A v = λ v, so compute A[3; 1; a] and equate components to λ[3; 1; a].

1) Relevant concepts
- An eigenvector v of matrix A satisfies A v = λ v for some scalar λ (the eigenvalue).
- This gives a system of equations obtained by equating each component of A v to λ times the corresponding component of v.
- For a nonzero eigenvector, at least one component of v must be nonzero (here v = [3, 1, a]^T has nonzero entries 3 and 1).

2) Compute A v (step-by-step)
A = [ [3, -1, 1],
      [-1, 3, 0],
      [1, 5, 1] ], v = [3, 1, a]^T.

First component:
(3)(3) + (-1)(1) + (1)(a) = 9 - 1 + a = 8 + a.

Second component:
(-1)(3) + (3)(1) + (0)(a) = -3 + 3 + 0 = 0.

Third component:
(1)(3) + (5)(1) + (1)(a) = 3 + 5 + a = 8 + a.

So A v = [8 + a, 0, 8 + a]^T.

3) Set up eigenvector equations and solve
Equate A v = λ v componentwise:
(1) 8 + a = λ * 3
(2) 0     = λ * 1
(3) 8 + a = λ * a

From (2): λ = 0.

Substitute λ = 0 into (1): 8 + a = 0 => a = -8.
Check (3): left 8 + a = 8 + (-8) = 0, right λ * a = 0 * (-8) = 0, so (3) holds. The vector [3, 1, -8] is nonzero, so valid.

4) Why this answer is correct
λ must be 0 because the second component of A v is 0 while the corresponding entry of v is 1, forcing λ = 0. That immediately forces 8 + a = 0, so a = -8. All three component equations are satisfied, confirming [3;1;-8] is an eigenvector with eigenvalue 0.

5) Why other possibilities fail
Any λ ≠ 0 contradicts equation (2) since 0 = λ·1 would be impossible. Any a ≠ -8 would make the first and third components 8 + a nonzero while λ = 0 forces those entries to be zero, so the eigenvector condition would fail.

6) Conclusion
Therefore a = -8 and the corresponding eigenvalue is λ = 0, matching the provided answers.

Question text 4Marks If [math: [31a]] is an eigenvector of [math: A=[3−11−130151],] then [math: a=] Answer 1[input], and the corresponding eigenvalue is [math: λ=] Answer 2[input].Notes Report question issue Question 3 Notes多项填空题

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