Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam solved
Math 225 Final Exam...
Math 225 Final Exam solved
Math 225 Final Exam solved
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.
, Math 225 Final Exam solved
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.
If AP=PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. -
CORRECT ANSWER The 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 ANSWER The 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 ANSWER No. 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.
How to calculate the distance between two vectors - CORRECT ANSWER (u-v) then
find the magnitude
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