MATH 225 FINAL EXAM QUIZZES FULL
ELABORATED WITH DETAILED CORRECT
SOLUTIONS
If the columns of A are linearly dependent - answer ✔✔-Then the matrix is not invertible and an
eigenvalue is 0
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.
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.
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
In a simplified n x n matrix the Eigenvalues are - answer ✔✔-The values of the main diagonal
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.
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
2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of the entries on the main diagonal of A.
, 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.
(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.
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.
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.
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.
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 false. Choose the correct answer
below. - answer ✔✔-The statement is true because it is the definition of the algebraic multiplicity of an
eigenvalue of A.
Determine whether the statement "A row replacement operation on A does not change the
eigenvalues" is true or false. Choose the correct answer below. - answer ✔✔-The statement is false
because row operations on a matrix usually change its eigenvalues.
A matrix A is diagonalizable if A has n eigenvectors. - answer ✔✔-The statement is false. A
diagonalizable matrix must have n linearly independent eigenvectors.
ELABORATED WITH DETAILED CORRECT
SOLUTIONS
If the columns of A are linearly dependent - answer ✔✔-Then the matrix is not invertible and an
eigenvalue is 0
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.
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.
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
In a simplified n x n matrix the Eigenvalues are - answer ✔✔-The values of the main diagonal
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.
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
2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of the entries on the main diagonal of A.
, 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.
(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.
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.
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.
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.
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 false. Choose the correct answer
below. - answer ✔✔-The statement is true because it is the definition of the algebraic multiplicity of an
eigenvalue of A.
Determine whether the statement "A row replacement operation on A does not change the
eigenvalues" is true or false. Choose the correct answer below. - answer ✔✔-The statement is false
because row operations on a matrix usually change its eigenvalues.
A matrix A is diagonalizable if A has n eigenvectors. - answer ✔✔-The statement is false. A
diagonalizable matrix must have n linearly independent eigenvectors.
