Math 225 Final Exam Latest update
2025/2026
If the columns of A are linearly dependent - correct answerThen 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? - correct answerLeft-
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. - correct answerIf
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 - correct answerAdd 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 - correct answerThe values of the main
diagonal
Use a property of determinants to show that A and AT have the same characteristic
polynomial - correct answerStart 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. - correct
answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe
statement is false because row operations on a matrix usually change its eigenvalues.
A matrix A is diagonalizable if A has n eigenvectors. - correct answerThe statement is
false. A diagonalizable matrix must have n linearly independent eigenvectors.
If A is diagonalizable, then A has n distinct eigenvalues - correct answerThe statement
is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n
linearly independent eigenvectors.
If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. -
correct answerThe statement is true. Let v be a nonzero column in P and let λ be the
corresponding diagonal element in D. Then AP=PD implies that Av=λv, which means
that v is an eigenvector of A.
If A is invertible, then A is diagonalizable. - correct answerThe statement is false. An
invertible matrix may have fewer than n linearly independent eigenvectors, making it not
diagonalizable.
A is a 3×3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A
diagonalizable? Why? - correct answerNo. The sum of the dimensions of the
eigenspaces equals 2 and the matrix has 3 columns. The sum of the dimensions of the
eigenspace and the number of columns must be equal.
2025/2026
If the columns of A are linearly dependent - correct answerThen 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? - correct answerLeft-
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. - correct answerIf
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 - correct answerAdd 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 - correct answerThe values of the main
diagonal
Use a property of determinants to show that A and AT have the same characteristic
polynomial - correct answerStart 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. - correct
answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe 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. - correct answerThe
statement is false because row operations on a matrix usually change its eigenvalues.
A matrix A is diagonalizable if A has n eigenvectors. - correct answerThe statement is
false. A diagonalizable matrix must have n linearly independent eigenvectors.
If A is diagonalizable, then A has n distinct eigenvalues - correct answerThe statement
is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n
linearly independent eigenvectors.
If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. -
correct answerThe statement is true. Let v be a nonzero column in P and let λ be the
corresponding diagonal element in D. Then AP=PD implies that Av=λv, which means
that v is an eigenvector of A.
If A is invertible, then A is diagonalizable. - correct answerThe statement is false. An
invertible matrix may have fewer than n linearly independent eigenvectors, making it not
diagonalizable.
A is a 3×3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A
diagonalizable? Why? - correct answerNo. The sum of the dimensions of the
eigenspaces equals 2 and the matrix has 3 columns. The sum of the dimensions of the
eigenspace and the number of columns must be equal.
