MATH 225 FINAL EXAMS WITH CORRECT
SOLUTIONS 2026/2027/UPDATED !!!
1. If the columns of A are linearly dependent - ANSWER-Then the matrix is not invertible
and an eigenvalue is 0
2. Note that A−1 exists. In order for λ−1 to be an eigenvalue of A−1, there must exist a
nonzero x such that Upper A Superscript negative 1 Baseline Bold x equals lambda
Superscript negative 1 Baseline Bold x . A−1x=λ−1x. Suppose a nonzero x satisfies
Ax=λx. What is the first operation that should be performed on Ax=λx so that an
equation similar to the one in the previous step can be obtained? - ANSWER-Left-
multiply both sides of Ax=λx by A−1.
3. Show that if A2 is the zero matrix, then the only eigenvalue of A is 0. - ANSWER-If
Ax=λx for some x≠0, then 0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must
be zero. Thus, each eigenvalue of A is zero.
4. Finding the characteristic polynomial of a 3 x 3 matrix - ANSWER-Add the first two
columns to the right side of the matrix and then add the down diagonals and subtract
the up diagonals
5. In a simplified n x n matrix the Eigenvalues are - ANSWER-The values of the main
diagonal
6. Use a property of determinants to show that A and AT have the same characteristic
polynomial - ANSWER-Start with detAT−λI)=detAT−λI)=det(A−λI)T. Then use the
formula det AT=det A.
, 7. The determinant of A is the product of the diagonal entries in A. Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWER-The statement is false because the determinant of the
8. 2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of the entries on the main
diagonal of A.
9. An elementary row operation on A does not change the determinant. Choose the
correct answer below. - ANSWER-The statement is false because scaling a row also
scales the determinant by the same scalar factor.
10. (det A)(det B)=detAB. Select the correct choice below and, if necessary, fill in the
answer box to complete your choice. - ANSWER-The statement is true because it is the
Multiplicative Property of determinants.
11. If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
Select the correct choice below and, if necessary, fill in the answer box to complete
your choice. - ANSWER-The statement is false because in order for 5 to be an
eigenvalue of A, the characteristic polynomial would need to have a factor of λ−5.
12. Determine whether the statement "If A is 3×3, with columns a1, a2, a3, then det A
equals the volume of the parallelepiped determined by a1, a2, a3" is true or false.
Choose the correct answer below. - ANSWER-The statement is false because det A
equals the volume of the parallelepiped determined by a1, a2, a3. It is possible that
det A≠det A.
13. Determine whether the statement "det AT=(−1)det A"is true or false. Choose the
correct answer below. - ANSWER-The statement is false because det AT=det A for any
n×n matrix A.
14. Determine whether the statement "The multiplicity of a root r of the characteristic
equation of A is called the algebraic multiplicity of r as an eigenvalue of A" is true or
SOLUTIONS 2026/2027/UPDATED !!!
1. If the columns of A are linearly dependent - ANSWER-Then the matrix is not invertible
and an eigenvalue is 0
2. Note that A−1 exists. In order for λ−1 to be an eigenvalue of A−1, there must exist a
nonzero x such that Upper A Superscript negative 1 Baseline Bold x equals lambda
Superscript negative 1 Baseline Bold x . A−1x=λ−1x. Suppose a nonzero x satisfies
Ax=λx. What is the first operation that should be performed on Ax=λx so that an
equation similar to the one in the previous step can be obtained? - ANSWER-Left-
multiply both sides of Ax=λx by A−1.
3. Show that if A2 is the zero matrix, then the only eigenvalue of A is 0. - ANSWER-If
Ax=λx for some x≠0, then 0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must
be zero. Thus, each eigenvalue of A is zero.
4. Finding the characteristic polynomial of a 3 x 3 matrix - ANSWER-Add the first two
columns to the right side of the matrix and then add the down diagonals and subtract
the up diagonals
5. In a simplified n x n matrix the Eigenvalues are - ANSWER-The values of the main
diagonal
6. Use a property of determinants to show that A and AT have the same characteristic
polynomial - ANSWER-Start with detAT−λI)=detAT−λI)=det(A−λI)T. Then use the
formula det AT=det A.
, 7. The determinant of A is the product of the diagonal entries in A. Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWER-The statement is false because the determinant of the
8. 2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of the entries on the main
diagonal of A.
9. An elementary row operation on A does not change the determinant. Choose the
correct answer below. - ANSWER-The statement is false because scaling a row also
scales the determinant by the same scalar factor.
10. (det A)(det B)=detAB. Select the correct choice below and, if necessary, fill in the
answer box to complete your choice. - ANSWER-The statement is true because it is the
Multiplicative Property of determinants.
11. If λ+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
Select the correct choice below and, if necessary, fill in the answer box to complete
your choice. - ANSWER-The statement is false because in order for 5 to be an
eigenvalue of A, the characteristic polynomial would need to have a factor of λ−5.
12. Determine whether the statement "If A is 3×3, with columns a1, a2, a3, then det A
equals the volume of the parallelepiped determined by a1, a2, a3" is true or false.
Choose the correct answer below. - ANSWER-The statement is false because det A
equals the volume of the parallelepiped determined by a1, a2, a3. It is possible that
det A≠det A.
13. Determine whether the statement "det AT=(−1)det A"is true or false. Choose the
correct answer below. - ANSWER-The statement is false because det AT=det A for any
n×n matrix A.
14. Determine whether the statement "The multiplicity of a root r of the characteristic
equation of A is called the algebraic multiplicity of r as an eigenvalue of A" is true or
