1|Page
MATH 225N Statistics Final Exam, Most Recent Exam
2026 Actual Complete Real Verified Exam Questions
And Correct Answers (Verified Answers) Already
Graded A+ Guaranteed Success!! Newest Exam!!
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.
,2|Page
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.
, 3|Page
(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
MATH 225N Statistics Final Exam, Most Recent Exam
2026 Actual Complete Real Verified Exam Questions
And Correct Answers (Verified Answers) Already
Graded A+ Guaranteed Success!! Newest Exam!!
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.
,2|Page
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.
, 3|Page
(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
