Lesson Worksheet: Eigenvalues and Eigenvectors for Special Matrices Mathematics

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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=−6200𝑚3005 be a 3×3 matrix, where 𝑚 is some real number. Suppose that the matrix 𝐴 has an eigenvalue 9.

Determine the value of 𝑚.

  • A5
  • B9
  • C−6
  • D1

Does the matrix 𝐴 have eigenvalues other than 9?

  • ANo
  • BYes

Q2:

Find all corresponding eigenvectors for the matrix 𝐴=⎡⎢⎢⎣10002100−23107651⎤⎥⎥⎦.

  • A⎡⎢⎢⎣0011⎤⎥⎥⎦
  • B⎡⎢⎢⎣1111⎤⎥⎥⎦
  • C⎡⎢⎢⎣0111⎤⎥⎥⎦
  • D⎡⎢⎢⎣0001⎤⎥⎥⎦

Q3:

Find all corresponding eigenvectors for the matrix 𝐴=1091.

  • A01
  • B−11
  • C21
  • D11

Q4:

Find all corresponding eigenvectors for the matrix 𝐴=100210−651.

  • A111
  • B011
  • C101
  • D001

Q5:

Let 𝐴=320𝑚 be a 2×2 matrix, where 𝑚 is some real number. Suppose that the matrix 𝐴 has an eigenvalue 8. Find all corresponding eigenvectors for the matrix 𝐴.

  • A10
  • B25,10
  • C25
  • D−52,10
  • E35,10

Q6:

Let 𝐴=𝑚−1302400−5 be a 3×3 matrix, where 𝑚 is some real number. Suppose that the matrix 𝐴 has an eigenvalue −2. Find all corresponding eigenvectors for the matrix 𝐴.

  • A−25−1221,100,−140
  • B−2512−21,100,−140
  • C251221,100,−140
  • D−25−1221,100,140
  • E−12−1221,100,−120

Q7:

Let the matrix 𝐴=123015001.

Find all the eigenvalues of the matrix 𝐴.

  • A𝜆=−1
  • B𝜆=0
  • C𝜆=3
  • D𝜆=1

Find all corresponding eigenvectors for the matrix 𝐴.

  • A301
  • B100
  • C−110
  • D210

Q8:

Let the matrix 𝐴=⎡⎢⎢⎣12310−11000230005⎤⎥⎥⎦.

Find all the eigenvalues of the matrix 𝐴.

  • A𝜆=1,𝜆=−1,𝜆=2,𝜆=5
  • B𝜆=−1,𝜆=1,𝜆=−2,𝜆=−5
  • C𝜆=−1,𝜆=2,𝜆=5
  • D𝜆=1,𝜆=2,𝜆=5

Find all corresponding eigenvectors for the matrix 𝐴.

  • A⎡⎢⎢⎣1321212⎤⎥⎥⎦,⎡⎢⎢⎣−11130⎤⎥⎥⎦,⎡⎢⎢⎣1100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • B⎡⎢⎢⎣1321212⎤⎥⎥⎦,⎡⎢⎢⎣11130⎤⎥⎥⎦,⎡⎢⎢⎣−1100⎤⎥⎥⎦
  • C⎡⎢⎢⎣1331212⎤⎥⎥⎦,⎡⎢⎢⎣11130⎤⎥⎥⎦,⎡⎢⎢⎣−1100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • D⎡⎢⎢⎣1321212⎤⎥⎥⎦,⎡⎢⎢⎣11130⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦

Q9:

Let matrix 𝐴=1201.

Find all the eigenvalues of matrix 𝐴.

  • A𝜆=2
  • B𝜆=1
  • C𝜆=−1
  • D𝜆=0

Find all the corresponding eigenvectors for matrix 𝐴.

  • A11
  • B21
  • C−11
  • D10

Q10:

Let 𝐴=⎡⎢⎢⎣𝑚123021−100350002⎤⎥⎥⎦ be a 4×4 matrix, where 𝑚 is some real number. Suppose that the matrix 𝐴 has an eigenvalue 1 . Find all corresponding eigenvectors for the matrix 𝐴.

  • A⎡⎢⎢⎣2320⎤⎥⎥⎦,⎡⎢⎢⎣1100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • B⎡⎢⎢⎣4220⎤⎥⎥⎦,⎡⎢⎢⎣2100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • C⎡⎢⎢⎣3−230⎤⎥⎥⎦,⎡⎢⎢⎣1100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • D⎡⎢⎢⎣3220⎤⎥⎥⎦,⎡⎢⎢⎣1−100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦
  • E⎡⎢⎢⎣3220⎤⎥⎥⎦,⎡⎢⎢⎣1100⎤⎥⎥⎦,⎡⎢⎢⎣1000⎤⎥⎥⎦

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