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Math 307
Spring, 2003
Hentzel
Time: 1:10-2:00 MWF
Room: 1324 Howe Hall
Office 432 Carver
Phone 515-294-8141
E-mail: [email protected]
http://www.math.iastate.edu/hentzel/class.307.ICN
Text: Linear Algebra With Applications,
Second Edition
Otto Bretscher
Friday, Feb 7 Chapter 2
No hand-in-homework assignment
Main Idea: I do not want any surprises on the
test.
Key Words: Practice test
Goal: Test over the material taught in class.
1. The function T|x| = |x-y| is a
|y| |y-x|
linear transformation.
True. It has matrix | 1 -1 |.
|-1 1 |
• 2. Matrix | 1/2 -1/2 | represents a
•
| 1/2 1/2 |
• rotation.
• False
(1/2)2 + (1/2)2 = 1/2 =/= 1
• 3. If A is any invertible nxn matrix, then
• rref(A) = In.
• True. A matrix is invertible if and only
• if its RCF is the identity.
• 4. The formula (A2)-1 = (A-1)2 holds
• for all invertible matrices A.
• True.
A A A-1 A-1 = I.
• 5. The formula AB=BA holds for all nxn
• matrices A and B.
• False.
•
| 0 1| |0 0| =/= | 0 0 | | 0 1 |
| 0 0| |1 0|
|1 0| |0 0|
• 6. If AB = In for two nxn matrices A and B,
• then A must be the inverse of B.
• True. This is false if A and B are not
• square.
• 7. If A is a 3x4 matrix and B is a 4x5
matrix, then AB will be a 5x3 matrix.
• False.
AB will be a 3x5 matrix.
• 8. The function T|x| = |y| is a linear
•
|y| |1|
• transformation.
• False.
•
T (2 |0|) = |0| =/= 2 T|0| = | 0 |
|0|
|1|
|0| | 2 |
• 9. The matrix | 5 6 | represents a
•
|-6 5 |
• rotation-dilation.
• True. The dilation is by Sqrt[61] the angle
• is ArcTan[-6/5] = -0.876058 radians
• 10. If A is any invertible
nxn matrix, then
• A commutes with A-1.
• True.
By definition, A A-1 =
A-1A = I
• 11. Matrix | 1 2 | is invertible.
•
| 3 6 |
• False. The RCF is | 1 2 |.
•
|0 0|
•
|1 1 1|
• Matrix | 1 0 1 | is invertible.
•
|1 1 0|
•
• True.
•
|1 1 1|
| 1 0 1| | 1 0 0 |
•
| 1 0 1 | ~ | 0 1 0| ~ | 0 1 0 |
•
|1 1 0|
| 0 1 -1| | 0 0 1 |
• 13. There is an upper triangular 2x2
• matrix A such that A2 = | 1 1 |
•
|0 1|
• True.
•
A = | 1 1/2 | is one possibility.
| 0 1 |
• 14. The function
• T|x| = |(y+1)2 – (y-1)2 | is a linear
• |y| |(x-3)2 – (x+3)2 |
• transformation.
• True. T|x| = | 4 y|.
•
|y|
|-12 x|
• 15. Matrix | k -2 | is invertible for all
•
| 5 k-6 |
• real numbers k.
• True.
• | k -2 | ~ | 1 (k-6)/5 | ~ | 1
(k-6)/5 |
• | 5 k-6 | | k
-2 | | 0 (-k^2+6k-10)/5|
• This polynomial has roots 3 (+/-) i so for all
REAL numbers k, the RCF is I and it is
invertible.
• 16. There is a real number k such that the
• matrix | k-1 -2 | fails to be invertible.
•
| -4 k-3 |
• True. k = -1 | -2 -2 | k = 5 | 4 -2 |.
•
| -4 -4 |
| -4 2 |
• 17. There is a real number k such that
• the matrix | k-2
3 | fails to be
•
| -3 k-2 |
• invertible.
•
•
•
•
•
False.
| k-2 3 | ~ | 1 -(k-2)/3 | ~ | 1 -(k-2)/3 |
| -3 k-2 | | k-2 3 | | 0 (k-2)2+3|
the roots are k = 2 (+/-) i Sqrt[3] which are
not real.
• 18. Matrix | -0.6 0.8 | represents a
•
|-0.8 -0.6 |
• rotation.
• True: theta = Pi + ArcCos[0.6] = 4.06889
• 19. The formula det(2A) = 2 det(A) holds
• for all 2x2 matrices A.
• False.
det(2A) = 4 det(A).
• 20. There is a matrix A such that
• | 1 2 | A | 5 6 | = | 1 1 |.
• |3 4| |78|
|1 1|
• True
•
| 1 2 | -1 | 1 1 | | 5 6 ||-1
| 3 4 |
|1 1| |7 8|
• Should work.
•
1/2 | 1 -1 |
| -1 1 |
• 21. There is a matrix A such that
• A | 1 1 | = | 1 2 |.
• | 1 1| | 1 2|
• False Any linear combination of the rows
• of | 1 1 | will look like | x x |.
•
|1 1|
| y y |
• 22. There is a matrix A such that
•
•
| 1 2 | A = | 1 1 |,
| 1 2 |
| 1 1 |
• True. | 1 1 | works.
•
|0 0|
• 23. Matrix | -1 2 | represents a shear.
•
| -2 3 |
• False
• | -1 2 | |x| = | -x + 2y| = |x| +2(-x+y) | 1|
• | -2 3 | |y|
| -2x+3y| |y|
| 1|
• The fixed vector has | 1 |.
•
|1|
•
• 24. | 1 k |3 = | 1 3k | for all real
•
|0 1 |
| 0 1|
• numbers k.
• True:
• 25. The matrix product
•
•
| a b | | d -b | is always a scalar
| c d | | -c a |
• of I2.
• True. The scalar is ad-bc.
• 26. There is a nonzero upper triangular
• 2x2 matrix A such that A2 = | 0 0 |.
•
| 0 0|
• True.
•
A = | 0 1 | is one possibility.
|0 0|
• 27. There is a positive integer n such that
•
•
| 0 -1 | n =
| 1 0|
I2.
• True. n = 4 is one possibility.
• 28. There is an invertible 2x2 matrix A
• such that A-1 = | 1 1 |.
•
|1 1|
• False. The RCF of | 1 1 | = | 1 1 |
•
| 1 1|
| 0 0|
• so | 1 1 | cannot be an invertible matrix.
•
| 1 1|
• 29. There is an invertible nxn matrix with
two identical rows.
• False. If A has two identical rows, then
• AB has 2 identical rows also. Thus
• AB cannot be I.
• 30. If A2 = In, then matrix A must be
invertible.
• True. In fact, A is its own inverse.
• 31. If A17 = I2, then A must be I2.
• False A = | Cos[t]
•
| Sin[t]
-Sin[t] |
Cos[t] |
• Where t = 2 Pi/17 should work.
• 32. If A2 = I2 , then A must be either I2 or –I2.
• False A = | -1 0 | is one possibility.
•
| 0 1|
• 33. If matrix A is invertible, then matrix
• 5 A is invertible as well.
• True.
And (5A)-1 = 1/5 A-1.
• 34.
If A and B are two 4x3 matrices such
• that AV = BV for all vectors v in R3, then
• matrices A and B must be equal.
• True. It follows that AI = BI for the 3x3
• identity matrix I. Thus A=B.
• 35. If matrices A and B commute, then the
• formula A2B = BA2 must hold.
• True.
A2B = AAB = ABA=BAA=BA2.
• 36. If A2 = A for an invertible nxn matrix
• A, then A must be In.
• True. Multiply through by A-1 giving A=I.
• 37. If matrices A and B are both invertible,
• then matrix A+B must be invertible as well.
• False. Let B = -A.
• 38. The equation A2 = A holds for all 2x2
• matrices A representing an orthogonal
• projection.
• True.
Once you have projected once by
• A, subequent actions by A will simply fix the
• vector.
•
• 39. If matrix | a b c | is invertible, then
•
|d e f |
•
|g h I |
• matrix | a b | must be invertible as well.
•
| d e |
•
• False.
•
•
| 0 0 1|
| 0 1 0 | Is an example.
| 1 0 0|
• 40. If A2 is invertible, then
• matrix A itself must be invertible.
• True.
For A2
to be defined, then
• A must be square.
If AAB = I, then
• A must be right invertible so A is
• invertible.
• 41. The equation A-1 = A holds for all 2x2
• matrices A representing a reflection.
• True. For a reflection A2 = I.
• 42. The formula (AV).(AW) = V.W holds
• for all invertible 2x2 matrices A and for
• all vectors V and W in R2.
• False.
•
| 1 1 | | 0 | .| 1 1 | | 1 | = 1
| 0 1 | | 1 | | 0 1| | 0 |
• 43. There exist a 2x3 matrix A and a 3x2
• matrix B such that AB = I3.
• True.
•
•
|1 0 0||1 0| = |1 0 |
| 0 1 0 | | 0 1|
| 0 1 |
| 0 0|
• 44. There exist a 3x2 matrix A and a 2x3
• matrix B such that AB = I3.
• False. There must be some X =/= 0
• such that BX = 0. Then 0 = ABX = X.
• Contradiction.
• 45. If A2 + 3A + 4 I3 = 0 for a 3x3 matrix
• A then A must be invertible.
• True.
A(A+3) = -4 I3
• so the inverse of A is (-1/4)(A+3).
• 46. If A is an nxn such that A2 = 0, then
• matrix In+A must be invertible.
• True.
(In+A)(In-A) = I.
• 47. If matrix A represents a shear, then
•
• the formula A2-2A+I2 = 0 must hold.
• True. (A-I)X will be a fixed vector.
• So A(A-I)X = (A-I)X which means
• A2-2A+I = 0.
•
48. If T is any linear transformation
• from R3 to R3, then T(VxW) = T(V)xT(W)
• for all vectors V and W in R3.
•
| 101|
|1|
|0|
• False. T = | 0 1 1 | V = | 0 | W = | 0 |
•
| 001|
|0|
|1|
•
•
| 0| | 0|
| 1| | 1|
| 0|
• T[VxW] = T| -1 | = |-1 | (TV)x(TW) = | 0 | x| 1 } = | -1 |.
•
| 0|
|0|
| 0| } 1}
| 1|
•
• 49. There is an invertible 10x10 matrix
• that has 92 ones among its entries.
•
•
•
•
False. There are only 8 entries which
are not one. At least 2 columns have
only ones. Matrices with 2 identical
columns are not invertible.
• 50. The formula rref(AB) = rref(A)rref(B)
• holds for all mxn matrices A and for all
• nxp matrices B.
• False A = B = | 0 0 |
•
| 1 0|
• rref(AB) =| 0 0 | rref(A)rref(B) = | 1 0 |
•
| 0 0 |
|0 0 |