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2 by 2 Eigenvalues and Eigenvectors worksheets

Solve exact designed 2 by 2 cases with distinct or repeated eigenvalues and valid eigenspace bases.

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Find the characteristic polynomial, exact eigenvalues, and a legitimate eigenvector basis for every eigenspace.

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Use the characteristic polynomial and null spaces to find eigenvalues and eigenvectors.

Worked example

Find eigenpairs of diag(2,5).

lambda=2 with span{(1,0)}; lambda=5 with span{(0,1)}.

Each diagonal entry scales its coordinate-axis basis vector.

Teaching notes

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  • Check every proposed vector with Av=lambda v.
  • Report an eigenspace basis rather than one arbitrary scaled copy.
  • Compare algebraic and geometric multiplicity in repeated cases.

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Math SheetNameScoreTeacherDate2 by 2 Eigenvalues and EigenvectorsFind the characteristic polynomial, exact eigenvalues, and a legitimate eigenvector basis for every eigenspace.1.For A = [[-1, 0], [0, -1]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.2.For A = [[-3, 0], [12, 3]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.Show your reasoning. Keep exact values unless instructed otherwise.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDate2 by 2 Eigenvalues and EigenvectorsFind the characteristic polynomial, exact eigenvalues, and a legitimate eigenvector basis for every eigenspace.3.For A = [[2, 8], [0, -2]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.4.For A = [[3, -2], [4, -3]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.Show your reasoning. Keep exact values unless instructed otherwise.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDate2 by 2 Eigenvalues and EigenvectorsFind the characteristic polynomial, exact eigenvalues, and a legitimate eigenvector basis for every eigenspace.5.For A = [[-6, 4], [-1, -2]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.6.For A = [[5, 5], [0, 0]], find p(lambda), the exact eigenvalues, and a basis for each eigenspace.Show your reasoning. Keep exact values unless instructed otherwise.math-sheet.comPage 3 of 3

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