Worksheet: Eigenvalues and Eigenvectors for Special Matrices

In this worksheet, we will practice finding the eigenvalues and eigenvectors of special matrices such as upper triangular, lower triangular, and diagonal matrices.

Q1:

Let 𝐴=ο˜π‘šβˆ’1302400βˆ’5 be a 3Γ—3 matrix, where π‘š is some real number. Suppose that the matrix 𝐴 has an eigenvalue βˆ’2. Find all corresponding eigenvectors for the matrix 𝐴.

  • Aο˜βˆ’25βˆ’1221,100,ο˜βˆ’140
  • Bο˜βˆ’2512βˆ’21,100,ο˜βˆ’140
  • C251221,100,ο˜βˆ’140
  • Dο˜βˆ’25βˆ’1221,100,140
  • Eο˜βˆ’12βˆ’1221,100,ο˜βˆ’120

Q2:

Find all corresponding eigenvectors for the matrix 𝐴=100210βˆ’651.

  • A111
  • B011
  • C101
  • D001

Q3:

Let 𝐴=ο˜βˆ’6200π‘š3005 be a 3Γ—3 matrix, where π‘š is some real number. Suppose that the matrix 𝐴 has an eigenvalue 9.

Determine the value of π‘š.

  • A5
  • B9
  • Cβˆ’6
  • D1

Does the matrix 𝐴 have eigenvalues other than 9?

  • ANo
  • BYes

Q4:

Find all corresponding eigenvectors for the matrix 𝐴=1091.

  • A01
  • Bο”βˆ’11
  • C21
  • D11

Q5:

Let 𝐴=320π‘šο  be a 2Γ—2 matrix, where π‘š is some real number. Suppose that the matrix 𝐴 has an eigenvalue 8. Find all corresponding eigenvectors for the matrix 𝐴.

  • A10
  • B25,10
  • C25
  • Dο”βˆ’52,10
  • E35,10

Q6:

Find all corresponding eigenvectors for the matrix 𝐴=⎑⎒⎒⎣10002100βˆ’23107651⎀βŽ₯βŽ₯⎦.

  • A⎑⎒⎒⎣0011⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣1111⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣0111⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣0001⎀βŽ₯βŽ₯⎦

Q7:

Let the matrix 𝐴=123015001.

Find all the eigenvalues of the matrix 𝐴.

  • Aπœ†=βˆ’1
  • Bπœ†=0
  • Cπœ†=3
  • Dπœ†=1

Find all corresponding eigenvectors for the matrix 𝐴.

  • A301
  • B100
  • Cο˜βˆ’110
  • D210

Q8:

Let the matrix 𝐴=⎑⎒⎒⎣12310βˆ’11000230005⎀βŽ₯βŽ₯⎦.

Find all the eigenvalues of the matrix 𝐴.

  • Aπœ†=1,πœ†=βˆ’1,πœ†=2,πœ†=5
  • Bπœ†=βˆ’1,πœ†=1,πœ†=βˆ’2,πœ†=βˆ’5
  • Cπœ†=βˆ’1,πœ†=2,πœ†=5
  • Dπœ†=1,πœ†=2,πœ†=5

Find all corresponding eigenvectors for the matrix 𝐴.

  • A⎑⎒⎒⎣1321212⎀βŽ₯βŽ₯⎦,βŽ‘βŽ’βŽ’βŽ£βˆ’11130⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣1321212⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣11130⎀βŽ₯βŽ₯⎦,βŽ‘βŽ’βŽ’βŽ£βˆ’1100⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣1331212⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣11130⎀βŽ₯βŽ₯⎦,βŽ‘βŽ’βŽ’βŽ£βˆ’1100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣1321212⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣11130⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦

Q9:

Let matrix 𝐴=1201.

Find all the eigenvalues of matrix 𝐴.

  • Aπœ†=2
  • Bπœ†=1
  • Cπœ†=βˆ’1
  • Dπœ†=0

Find all the corresponding eigenvectors for matrix 𝐴.

  • A11
  • B21
  • Cο”βˆ’11
  • D10

Q10:

Let 𝐴=βŽ‘βŽ’βŽ’βŽ£π‘š123021βˆ’100350002⎀βŽ₯βŽ₯⎦ be a 4Γ—4 matrix, where π‘š is some real number. Suppose that the matrix 𝐴 has an eigenvalue 1 . Find all corresponding eigenvectors for the matrix 𝐴.

  • A⎑⎒⎒⎣2320⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣4220⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣2100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣3βˆ’230⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣3220⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1βˆ’100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎣3220⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1100⎀βŽ₯βŽ₯⎦,⎑⎒⎒⎣1000⎀βŽ₯βŽ₯⎦

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