Day 26 & 27 – Some Practice Problems for the Final Exam
 


3
1 2 2
1. Is v = 2 an eigenvector of A = 3 −2 1? Why or why not?
1
0 1 1
(A) Yes, v is an evect of A corresponding to eval λ =
(B) No, because
 


−1
3 0 −1
2. Is v =  2  an eigenvector of A =  2 3 1 ? Why or why not?
1
−3 4 5
(A) Yes, v is an evect of A corresponding to eval λ =
(B) No, because


3 6 7
3. Let A = 3 3 7.
5 6 5
Then an eigenvector of A corresponding to λ = −2 is v =
1
4. Which of the following must be true TRUE about an n × n diagonalizable matrix A?
(A) A is similar to a diagonal matrix
(B) A is row equivalent to a diagonal matrix
(C) A has the same eigenvalues as a diagonal matrix
(D) A has n distinct eigenvalues
(E) A has n linearly independent eigenvectors
(F) A is invertible
(G) A is singular
(H) A has full-rank (i.e. rank(A) = n)
(I) A is orthogonally diagonalizable
(J) The columns of A span Rn
5. Which of the following matrix must always be diagonalizable?
(A) A symmetric 7 × 7 matrix
(B) An invertible 7 × 7 matrix
(C) A 7 × 7 matrix whose evals are −1, −2, −3, 0, 1, 2, 3
(D) A 7 × 7 matrix with one eval with algebraic multiplicity 2 and 5
other evals with algebraic multiplicity 1
(E) A 7 × 7 matrix with one eigenspace which is 3-dimensional and
two other eigenspaces which is 2-dimensional each
(F) A 7×7 matrix whose characteristic polynomial is λ3 (λ−1)2 (λ+1)2
(G) A full-rank 7 × 7 matrix
2


( 1 2 )
2
6. Let y = −2 and W = Span 0 , 1 . Then ŷ, the orthogonal
1
1
1
projection of y onto W is
 
5/2
(A) 1/2
2


1
(B) −1
2
 
3
(C) 1
2
3


1/2
(D) −1/2
1
(E) N one of them
7. Suppose the following
= eig(A) command in

0 1 −1
1 0 −1
P =
1 −1 0
1 1
1
results were obtained from using the
MatLab:


1
1
0
0


−1
0 −1/2
0
D=
0
1
0
−1/3
0
0
0
0
What are the eigenvalues and eigenvectors of A?
Is A symmetric? Why or why not?
Find lim Ak
k→∞
4
>>[P,D]

0
0 

0 
1/4
8. Which of the following is/are TRUE about two similar matrices A
and B?
(A) They have the same eigenvalue(s)
(B) They have the same null-space
(C) They have the same row-space
(D) They have the same column-space
(E) They have the same rank
(F) They have the same determinant
(G) They have the same trace (note: the trace of a matrix is the sum
of its diagonal entries)
(H) The linear transformations that use A and B as standard matrix
have the same range
(I) The linear transformations that use A and B as standard matrix
have the same kernel
(J) An and B n also are similar, for every positive integer n
(K) They have the same diagonalizability (i.e. either both are diagonalizable or neither is diagonalizable)
5
9. Which of the following is/are TRUE about two row-equivalent matrices A and B?
(A) They have the same eigenvalue(s)
(B) They have the same null-space
(C) They have the same row-space
(D) They have the same column-space
(E) They have the same rank
(F) They have the same determinant
(G) They have the same trace (note: the trace of a matrix is the sum
of its diagonal entries)
(H) The linear transformations that use A and B as standard matrix
have the same range
(I) The linear transformations that use A and B as standard matrix
have the same kernel
(J) An and B n also are row-equivalent, for every positive integer n
(K) They have the same diagonalizability (i.e. either both are diagonalizable or neither is diagonalizable)
(The last two items are not covered in our class, see if you can figure
them out)
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10. Let u and v be vectors in R3 such that kuk = 2, kvk = 3, and cos θ =
1
− √ . Here, 0 ≤ θ ≤ π is the angle between u and v.
3
Evaluate the following expressions:
(a) The magnitudes ku + vk and ku − vk
(b) The dot product (2u + 3v) · (3u − 2v)
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

4
11. Let A = a1 a2 a3 . Given that −3 ∈ N ul(A). Which of the
1
following can be deduced from this fact?
(A) det(A) = 0
(B) The homogeneous equation Ax = 0 has trivial solution
(C) λ = 0 is an eigenvalue of A
(D) 4a1 − 3a2 = a3
(E) There exists a vector b ∈ R3 that does not belong to the column
space of A
(F) rank(A) = 1
(G) nullity(A) = 1
(H) A has only two pivots
(I) The linear transformation T (x) = Ax is a bijection
12. Let A be an n × n orthogonal matrix. Which of the following is/are
FALSE?
(A) If A = QR is the QR-decomposition of A then A = Q
(B) The columns of A are linearly independent
(C) Ax = x for all x ∈ Rn
(D) kAxk = kxk for all x ∈ Rn
(E) (Ax) · (Ay) = x · y for all x, y ∈ Rn
(F) AAT = In (In is the n × n identity matrix)
(G) AT A = In (In is the n × n identity matrix)
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13. Let M3 be the set of all 3 × 3 matrices with real-valued entries. Apply
the subspace test on each the following subsets of M3 to test whether
they are subspaces of M3 .
Subset
Contain zero Closed under
vector?
addition
The set of all 3 × 3 symmetric matrices
The set of all 3 × 3 invertible matrices
The set of all 3 × 3 rational matrices
(i.e. they have rational entries)
The set of all 3 × 3 rank-deficient matrices
(i.e. their rank is < 3)
The set of all 3 × 3 diagonalizable matrices
The set of all 3 × 3 orthogonal matrices
(i.e. they have orthonormal columns)
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Closed under
scalar-mult?
14. Let T : R3 → R3 be a linear transformation given by T (x) = Ax.
Suppose that T is also onto. Which of the following is/are FALSE?
(A) For each b ∈ R3 , there is an x ∈ R3 such that T (x) = b.
( 0 )
(B) Ker(T ) =
0
0
(C) The eigenvalues of A are non-zero
(D) The range and the domain are the same in this case
(E) The matrix A is diagonalizable
(F) The matrix A is invertible
(G) The columns of A are orthogonal
(H) A has a pivot on every column
(I) Ker(T ) is a subspace of the codomain in this case
(J) If v, w are distinct vectors in R3 then T (v) 6= T (w)
(K) Each b ∈ R2 must have a pre-image x ∈ R2
(L) Each x ∈ R2 must have an image b ∈ R2
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