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Math 225 Final Exam with correct answers.
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Math 225 Final Exam with correctanswers
If the columns of A are linearly dependent - correct 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? - correct 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. - correct 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 - correct 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 - correct answer -The values of the main diagonal
Use a property of determinants to show that A and AT have the same characteristic polynomial - correct
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. - correct 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.
- correct 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. - correct 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. - correct 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. - correct
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.
- correct 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. - correct 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. - correct 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. - correct answer -The statement is false. A
diagonalizable matrix must have n linearly independent eigenvectors.
If A is diagonalizable, then A has n distinct eigenvalues - correct answer -The statement is false. A
diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent
eigenvectors.
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