# 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 , and 1 with corresponding eigenvector . The matrix is defective.
• BThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
• CThe eigenvalues are 10 with corresponding eigenvector , and 1 with corresponding eigenvector . The matrix is defective.
• DThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
• EThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . 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 and 6 with corresponding eigenvector .
• BThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
• CThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
• DThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
• E The eigenvalues are with corresponding eigenvector and with corresponding eigenvector .

Q3:

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

• AThe eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and 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 defective.
• C The eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is defective.
• DThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.
• EThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . 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
• B
• C
• D
• E

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 and are real numbers.

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

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 , defective
• B , defective
• C , defective
• D , defective
• E , 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 with corresponding eigenvector and 1 with corresponding eigenvector .
• B The eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
• CThe eigenvalues are