BitWithBite
Mathematics Worksheet
🎯 Eigenvalues
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After this worksheet you can
Find eigenvalues of diagonal and triangular matrices.Use the characteristic equation.Relate the trace to the sum of eigenvalues.Relate the determinant to the product of eigenvalues.📚 Quick Recap
An eigenvalue tells you how much a special direction gets stretched or shrunk when a matrix transforms it — these special values unlock deep insight into a matrix's behavior.
🧠 Section A · Concept Check ● BEGINNER 4 × 1 = 4
1The eigenvalues of [[7,0],[0,1]] are:
2The trace and determinant of [[5,2],[2,5]] are:
3From λ2 − 10λ + 21 = 0, the eigenvalues of [[5,2],[2,5]] are:
4The eigenvalues of the upper-triangular [[2,1],[0,2]] are:
🧮 Section B · Problem Solving ● INTERMEDIATE 2 + 3×3 = 11
5The eigenvalues of a diagonal matrix are its entries.
6The sum of the eigenvalues equals the .
7Find the eigenvalues of [[7,0],[0,1]].
8Find the trace and determinant of [[5,2],[2,5]].
9Using the previous result, find the eigenvalues.
🚀 Section C · Challenge ● CHALLENGE 5
10Find the eigenvalues of [[2,1],[0,2]] (upper triangular).
💭 Reflection — the most useful thing I learned:
A ___/4 B ___/11 C ___/5 Total ___/20
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1-A 2-A 3-A 4-A | 5 diagonal 6 trace 7 = 7 and 1 8 = Trace 10, Determinant 21 9 = λ = 3 or 7 | 10 = 2 and 2 (repeated)