BitWithBite
Mathematics Worksheet
➡️ Eigenvectors
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After this worksheet you can
Find an eigenvector by solving (A − λI)v = 0.Recognise scalar multiples of eigenvectors.Understand why (A − λI) must be singular.Connect eigenvectors to diagonalizability.📚 Quick Recap
Once you know a matrix's eigenvalues, eigenvectors show you the exact directions that only get stretched or shrunk — never rotated — when the matrix acts on them.
🧠 Section A · Concept Check ● BEGINNER 4 × 1 = 4
1For [[5,2],[2,5]], λ = 3, A−3I = [[2,2],[2,2]]. An eigenvector is:
2For [[5,2],[2,5]], λ = 7, A−7I = [[−2,2],[2,−2]]. An eigenvector is:
3For [[2,1],[0,2]], λ = 2, A−2I = [[0,1],[0,0]]. An eigenvector is:
4If (1,−1) is an eigenvector, is (5,−5) also one?
🧮 Section B · Problem Solving ● INTERMEDIATE 2 + 3×3 = 11
5An eigenvector satisfies Av = .
6To find eigenvectors for λ, solve (A − )v = 0.
7For [[5,2],[2,5]] with eigenvalue 3, form (A−3I) and solve for the eigenvector.
8For [[5,2],[2,5]] with eigenvalue 7, form (A−7I) and solve for the eigenvector.
9For [[2,1],[0,2]] with eigenvalue 2, form (A−2I) and solve for the eigenvector.
🚀 Section C · Challenge ● CHALLENGE 5
10Is (5,−5) a valid eigenvector if (1,−1) is one for the same eigenvalue?
💭 Reflection — the most useful thing I learned:
A ___/4 B ___/11 C ___/5 Total ___/20
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✂ answer key — fold or cut before handing out
1-A 2-A 3-A 4-A | 5 λv 6 λI 7 = x+y=0, eigenvector (1,−1) 8 = −x+y=0, eigenvector (1,1) 9 = y=0, eigenvector (1,0) | 10 = Yes, it's a scalar multiple (5 times)