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Lesson: Defective and Nondefective Matrices

Worksheet • 16 Questions

Q1:

For the matrix find the eigenvalues and eigenvectors and determine whether it is defective.

  • A , defective
  • B , defective
  • C , defective
  • D , defective
  • E , defective

Q2:

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

  • AThe eigenvalues are 1 with corresponding eigenvector 3 1 1 , 2 with corresponding eigenvector 1 2 1 , and 3 with corresponding eigenvector 2 1 1 . The matrix is not defective.
  • BThe eigenvalues are 1 with corresponding eigenvector 3 2 2 , 2 with corresponding eigenvector 7 1 3 , and 3 with corresponding eigenvector 1 9 8 1 0 . The matrix is not defective.
  • CThe eigenvalues are 1 with corresponding eigenvector 4 2 2 , 2 with corresponding eigenvector 7 3 3 , and 3 with corresponding eigenvector 1 9 8 7 . The matrix is not defective.
  • DThe eigenvalues are 10 with corresponding eigenvector 1 3 2 2 , and 1 with corresponding eigenvector 7 2 3 . The matrix is defective.
  • EThe eigenvalues are 10 with corresponding eigenvector 8 2 1 , and 1 with corresponding eigenvector 3 1 1 . The matrix is defective.

Q3:

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

  • AThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.
  • BThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.
  • CThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is defective.
  • D The eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is defective.
  • EThe eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is not defective.

Q4:

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

  • AThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
  • B The eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
  • CThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
  • DThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
  • EThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .

Q5:

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

  • AThe eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
  • B The eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
  • C The eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
  • DThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
  • E The eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .

Q6:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • AA basis for the eigenspace associated with the lone eigenvalue 0 is 3 4 1 4 1 . The matrix is therefore defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 1 is 7 8 1 4 1 . The matrix is therefore defective.
  • CA basis for the eigenspace associated with the lone eigenvalue 0 is 3 4 3 2 1 . The matrix is therefore defective.
  • DA basis for the eigenspace associated with the eigenvalue 2 is 1 1 1 , a basis for the eigenspace associated with the eigenvalue 3 is 9 8 2 1 , and a basis for the eigenspace associated with the eigenvalue 6 is 0 1 1 . The matrix is therefore not defective.
  • EA basis for the eigenspace associated with the eigenvalue 2 is 1 1 1 , a basis for the eigenspace associated with the eigenvalue 3 is 9 8 2 1 , and a basis for the eigenspace associated with the eigenvalue 6 is 0 1 1 . The matrix is therefore not defective.

Q7:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 6 , 2 + 6 𝑖 , and 2 6 𝑖 are 1 1 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 6 , 2 + 6 𝑖 , and 2 6 𝑖 are 1 1 1 , 1 1 6 5 3 5 𝑖 , and 1 1 6 5 3 5 𝑖 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 6 , 2 + 6 𝑖 , and 2 6 𝑖 are 1 1 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 6 , 2 + 6 𝑖 , and 2 6 𝑖 are 1 1 1 , 1 1 6 5 3 5 𝑖 , and 1 1 6 5 3 5 𝑖 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 6 , 2 + 6 𝑖 , and 2 6 𝑖 are 7 1 7 1 9 1 7 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.

Q8:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • AA basis for the eigenspace associated with the lone eigenvalue 1 is 3 4 1 4 1 . The matrix is therefore defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 0 is 7 8 1 4 1 . The matrix is therefore defective.
  • CA basis for the eigenspace associated with the lone eigenvalue 1 is 3 4 3 2 1 . The matrix is therefore defective.
  • DA basis for the eigenspace associated with the eigenvalue 7 is 0 1 1 , and a basis for the eigenspace associated with the eigenvalue 2 is 9 8 2 1 . The matrix is therefore not defective.
  • EA basis for the eigenspace associated with the eigenvalue 7 is 0 1 1 , and a basis for the eigenspace associated with the eigenvalue 2 is 9 8 2 1 . The matrix is therefore not defective.

Q9:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 2 and 6 are 1 1 0 , 0 0 1 , and 1 1 1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 2 , 6 , and 4 are 1 1 0 , 1 1 1 , and 0 0 1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 2 and 4 are 1 1 0 , 0 0 1 , and 0 0 1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 2 and 4 are 1 1 0 , 0 0 1 , and 1 4 1 1 4 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 2 and 6 are 1 1 0 , 0 0 1 , and 1 5 1 1 5 respectively. The matrix is therefore not defective.

Q10:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 6 and 12 are 2 1 0 , 1 0 1 , and 1 0 1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 6 and 1 2 are 2 1 0 , 1 0 1 , and 1 7 0 1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 6 and 9 are 2 1 0 , 1 0 1 , and 0 0 0 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 6 and 1 2 are 1 5 0 1 , and 1 7 0 1 respectively. The matrix is therefore defective.
  • EBases for the eigenspaces associated with the eigenvalues 6 and 9 are 1 5 0 1 , and 1 6 0 1 respectively. The matrix is therefore defective.

Q11:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 1 and 2 are 1 1 0 , 1 0 1 and 1 2 1 1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 1 and 2 are 1 1 0 , 1 0 1 and 1 2 0 1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 1 and 2 are 1 1 1 and 1 2 1 1 respectively. The matrix is therefore defective.
  • DBases for the eigenspaces associated with the eigenvalues 1 and 2 are 1 1 2 , 1 0 1 and 1 2 1 1 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 1 and 2 are 1 1 2 and 1 2 1 1 respectively. The matrix is therefore defective.

Q12:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 4, 2 + 2 𝑖 , and 2 2 𝑖 are 1 1 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 4 , 2 + 2 𝑖 , and 2 2 𝑖 are 2 1 3 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 4, 1 + 𝑖 , and 1 𝑖 are 1 1 1 , 1 2 𝑖 1 𝑖 , and 1 2 𝑖 1 + 𝑖 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 4 , 1 + 𝑖 , and 1 𝑖 are 2 1 3 1 , 1 2 𝑖 1 𝑖 , and 1 2 𝑖 1 + 𝑖 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 4 , 2 + 2 𝑖 , and 2 2 𝑖 are 2 1 3 1 , 𝑖 𝑖 1 , and 𝑖 𝑖 1 respectively. The matrix is therefore not defective.

Q13:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 0, 18, and 1 2 are 1 3 2 3 1 , 1 0 1 , and 2 1 0 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 0, 13, and 1 7 are 1 3 2 3 1 , 7 1 4 5 4 , and 5 2 1 4 1 3 1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 0, 13, and 1 7 are 1 3 2 3 1 , 7 1 4 5 4 , and 5 2 1 4 1 3 1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 0, 18, and 1 2 are 1 3 2 3 1 , 1 0 1 , and 2 1 0 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 0 and 4 are 1 3 7 9 1 , 1 1 1 9 1 respectively. The matrix is therefore defective.

Q14:

Find the eigenvectors of the matrix

  • A 0 𝑖 1 𝑎 𝑖 𝑏 , 0 𝑖 1 𝑎 + 𝑖 𝑏 , 1 0 0 𝑐
  • B 0 𝑖 1 𝑎 𝑖 𝑏 , 0 𝑖 1 𝑎 + 𝑖 𝑏 , 1 0 0 𝑐
  • C 0 𝑖 1 𝑎 𝑖 𝑏 , 0 𝑖 1 𝑎 + 𝑖 𝑏 , 1 0 0 𝑐
  • D 0 𝑖 1 𝑎 + 𝑖 𝑏 , 0 𝑖 1 𝑎 𝑖 𝑏 , 1 0 0 𝑐
  • E 0 𝑖 1 𝑎 + 𝑖 𝑏 , 0 𝑖 1 𝑎 𝑖 𝑏 , 1 0 0 𝑐

Q15:

Let 𝑇 be the linear transformation that rotates all vectors in 2 counterclockwise through an angle of 𝜋 2 . Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A 𝑇 = 0 1 1 0 . Its eigenvalues are 𝑖 with corresponding eigenvector 𝑖 1 and 𝑖 with corresponding eigenvector 𝑖 1 .
  • B 𝑇 = 1 0 0 1 . Its eigenvalues are 1 with corresponding eigenvector 0 1 and 1 with corresponding eigenvector 1 1 .
  • C 𝑇 = 1 0 0 1 . Its eigenvalues are 1 with corresponding eigenvector 0 1 and 1 with corresponding eigenvector 1 0 .
  • D 𝑇 = 0 1 1 0 . Its eigenvalues are 𝑖 with corresponding eigenvector 𝑖 1 and 1 with corresponding eigenvector 1 1 .
  • E 𝑇 = 0 1 2 1 . Its eigenvalues are 2 with corresponding eigenvector 2 1 and 4 with corresponding eigenvector 1 1 .

Q16:

Find the eigenvalues and eigenvectors of the matrix where and are real numbers.

  • AIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
  • BIts eigenvalues are 1 with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
  • CIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
  • DIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
  • EIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
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