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Worksheet: Defective and Nondefective Matrices

Q1:

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

  • AThe eigenvalues are 10 with corresponding eigenvector 8 2 1 , and 1 with corresponding eigenvector 3 1 1 . The matrix is defective.
  • BThe 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.
  • CThe eigenvalues are 10 with corresponding eigenvector 1 3 2 2 , and 1 with corresponding eigenvector 7 2 3 . The matrix is defective.
  • DThe 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.
  • EThe 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.

Q2:

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

  • AThe eigenvalues are 3 with corresponding eigenvector 2 4 1 and 6 with corresponding eigenvector 1 8 2 .
  • BThe eigenvalues are 3 with corresponding eigenvector 1 2 2 and 6 with corresponding eigenvector 2 1 0 .
  • CThe eigenvalues are 3 with corresponding eigenvector 1 2 6 and 6 with corresponding eigenvector 0 0 2 .
  • DThe eigenvalues are 3 with corresponding eigenvector 1 2 2 and 6 with corresponding eigenvector 0 0 1 .
  • E The eigenvalues are 6 with corresponding eigenvector 2 5 4 and 3 with corresponding eigenvector 5 4 1 .

Q3:

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

  • AThe eigenvalues are 1 with corresponding eigenvector 1 3 7 9 1 , 3 with corresponding eigenvector 3 2 3 2 1 , and 2 with corresponding eigenvector 3 2 1 1 . The matrix is not defective.
  • BThe eigenvalues are 1 with corresponding eigenvector 7 8 1 8 1 , 3 with corresponding eigenvector 6 1 1 6 1 1 1 , and 2 with corresponding eigenvector 1 7 6 7 1 . The matrix is defective.
  • C The eigenvalues are 1 with corresponding eigenvector 3 4 1 4 1 , 3 with corresponding eigenvector 9 1 3 3 1 3 1 , and 2 with corresponding eigenvector 1 2 1 1 . The matrix is defective.
  • DThe eigenvalues are 1 with corresponding eigenvector 3 4 1 4 1 , 3 with corresponding eigenvector 9 1 3 3 1 3 1 , and 2 with corresponding eigenvector 1 2 1 1 . The matrix is not defective.
  • EThe eigenvalues are 1 with corresponding eigenvector 1 3 7 9 1 , 3 with corresponding eigenvector 3 2 3 2 1 , and 2 with corresponding eigenvector 3 2 1 1 . The matrix is not defective.

Q4:

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 and are , respectively. The matrix is therefore defective.
  • BBases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.

Q5:

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

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 eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
  • CA basis for the eigenspace associated with the eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
  • DA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
  • EA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore 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 , , and are , , and respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues , , and are , , and 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 eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
  • CA basis for the eigenspace associated with the eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
  • DA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
  • EA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore 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 and are , and respectively. The matrix is therefore defective.
  • BBases for the eigenspaces associated with the eigenvalues 6 and 9 are , and respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore defective.
  • DBases for the eigenspaces associated with the eigenvalues 6 and 12 are , and respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 6 and are , and 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 2 and 6 are , and respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 2 and 4 are , and respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

Q11:

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

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

Q12:

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

  • AIts eigenvalues are 𝑐 with corresponding eigenvector 1 0 0 , 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 , and 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 .
  • BIts eigenvalues are 𝑐 with corresponding eigenvector 1 0 0 , 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 , and 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 .
  • CIts eigenvalues are 𝑐 with corresponding eigenvector 1 0 0 , 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 , and 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 .
  • DIts eigenvalues are 𝑐 with corresponding eigenvector 1 0 0 , 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 , and 𝑖 𝑏 with corresponding eigenvector 0 𝑖 1 .
  • EIts eigenvalues are 1 with corresponding eigenvector 𝑐 0 0 , 𝑖 with corresponding eigenvector 0 𝑖 𝑏 𝑏 , and 𝑖 with corresponding eigenvector 0 𝑖 𝑏 𝑏 .

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

Q14:

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

  • A 5 4 3 1 , 7 8 3 1 , defective
  • B 1 1 3 1 , 1 2 1 1 , defective
  • C 4 1 1 1 , 4 3 2 1 , defective
  • D 3 1 1 1 , 1 2 1 1 , defective
  • E 1 1 4 1 , 4 3 2 1 , defective

Q15:

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 , , and are , , and respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

Q16:

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

  • A The eigenvalues are 2 with corresponding eigenvector 1 2 1 and 1 with corresponding eigenvector 1 1 1 .
  • B The eigenvalues are 2 with corresponding eigenvector 1 2 1 and 1 with corresponding eigenvector 1 1 1 .
  • CThe eigenvalues are