linear algebra with applications by otto bretscher. page 286. 1. the determinant of any diagonal nxn...
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Linear Algebra With Applications by Otto Bretscher.
Page 286.
1. The Determinant of any diagonal nxn matrix is the product of its diagonal entries.
True.
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2. If matrix B is obtained by swapping two rows of an nxn matrix A, then the equation
det(B) = -det(A) must hold.
True. Interchanging two rows changes the
sign of the determinant
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3. If A = [U V W] is any 3x3 matrix, then
det(A) = uo(vxw)
True. Just compare the two expressions.
both are simply the determinant of A.
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4. Det[ 4 A ] = 4 Det[ A ] for all 4x4 matrices A.
False. Det[4 A] = 4 4 Det[A] since each
row of 4 A is multiplied by 4.
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5. Det [ A+B ] = Det [ A ] + Det [ B ] for all 5x5 matrices A and B.
False. There is nothing known about the
determinant of the sum.
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6. The equation Det[ -A ] = Det[ A ] holds for all 6x6 matrices.
True. Each row has a sign change so the
determinant changes sign six times.
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7. If all the entries of a 7x7 matrix A are 7,
then Det [ A ] must be 7 7.
False. The matrix has identical rows so the
determinant is zero.
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8. An 8 x 8 matrix fails to be invertible if (and only if) its determinant is nonzero.
False. A matrix fails to be invertible if (and only if) its determinant is zero.
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9. If B is obtained by multiplying a column A by 9, then the equation det(B) = 9 det(A) must hold.
True. Multiplying a column by c multiplies the
determinant by c.
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10. Det (A10) = (Det A) 10 for all 10x10 matrices A.
True. Det (AB) = Det (A) Det (B).
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11. If two n x n matrices A and B are similar, then the equation Det ( A ) = Det ( B ) must hold.
True. Det ( A -1 B A)
= Det (A -1) Det (B) Det (A)
= Det (A -1 A) Det (B)
= Det (B).
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12. The determinant of all orthogonal matrices is 1.
False. It is either 1 or -1.
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13. If A is any n x n matrix, then
Det( A A T) = Det( A T A )
True. Both equal Det(A) 2
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14. There is an invertible matrix of the form
| a e f j |
| b 0 g 0 |
| c 0 h 0 |
| d 0 i 0 |
False. The determinant is zero so it cannot be
invertible.
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15. The matrix is invertible for all positive
constants k.
| k 2 1 4 |
| k -1 -2 |
| 1 1 1 |
True. The determinant is a degree 2
polynomial with roots k = -2 and k = -1. Thus it has no positive roots and is always non zero for positive k.
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16.
| 0 1 0 0 |
Det | 0 0 1 0 | = 1
| 0 0 0 1 |
| 1 0 0 0 |
False. Three row operations give the
identity. There are three sign changes.
The Determinant is -1.
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17. Matrix is invertible
| 9 100 3 7 |
| 5 4 100 8 |
| 100 9 8 7 |
| 6 5 4 100 |
True. The determinant is 97763383
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18 If A is an invertible nxn matrix, then Det(AT)
must equal Det(A -1 ).
False. Det(A T) = Det(A) = 1/Det(A -1 )
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19. If the determinant of a 4x4 matrix A is 4, then its rank must be 4.
True. If the rank were not 4, the determinant would be zero.
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20. There is a nonzero 4x4 matrix A such that Det (A) = Det (4 A).
True. A is not zero, but Det (A) does equal 0.
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21. If all the columns of a square matrix A are unit vectors, then the determinant of A must be less than or equal to 1.
True: | A X | = | x1 C1 + x2 C2+ … xn Cn|
<= |x1||C1|+|x2||C2| + ….+|xn||Cn|
<= |x1|+|x2| + …+|xn| = 1.
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22. If A is any noninvertible square matrix, then Det (A) = Det (rref(A).
True. Det (A) = 0. Det (rref(A)) = 0
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23. If the determinant of a square matrix is -1, then A must be an orthogonal matrix.
False. | 1 1 | is not orthogonal.
| 0 -1 |
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24. If all the entries of an invertible matrix A
are integers, then the entries of A -1 must be integers as well.
False. | 2 0 | -1 = | ½ 0 |
| 0 2 | | 0 ½ |
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25. There is a 4x4 matrix A whose entries are all 1 or -1 and such that Det (A) = 16.
True. | 1 1 1 1 |
| 1 1 -1 -1 |
| 1 -1 1 -1 |
| 1 -1 -1 1 |
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26. If the determinant of a 2x2 matrix A is 4, then the inequality | A v | <= 4 | v | must hold
for all vectors v in R 2.
False. | 2 100 | | 0 | = | 100 |
| 0 2 | | 1 | | 2 |
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27. If A = [ u,v,w] is a 3x3 matrix, then the formula det (A) = vo(uxw) must hold.
False. It is the opposite sign.
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28. There are invertible 2x2 matrices A and B such that Det [A+B] = Det [A]+Det [B].
True. | 1 0 | | 0 1 |
| 0 1 | | 1 0 |
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29. If all the entries of a square matrix are 1 or 0, then Det (A) must be 1,0, or -1.
| 0 1 1 |
False. Det | 1 0 1 | = 2
| 1 1 0 |
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30. If all the entries of a square matrix A are integers and Det [A] = 1, then the entries of
matrix A -1 must be integers as well.
True. A -1 = 1/Det(A) Adj(A)
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31, If A is any symmetric matrix, then
Det [A] = 1 or Det [A] = -1.
False Det | 0 2 | = -4
| 2 0 |
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32. If A is any skew-symmetric 4x4 matrix, then Det (A) = 0.
| 0 1 0 0 |
| -1 0 0 0 |
| 0 0 0 -1 |
| 0 0 1 0 |
has determinant equal to 1.
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+33. If Det [A] = Det [B] for two nxn matrices A
and B, then A and B must be similar.
False. | 1 0 | is not similar to | 1 1 |
| 0 1 | | 0 1 |
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34. Suppose A is an n x n matrix and B is obtained from A by swapping two rows of A. If Det [B] < Det [A], then A must be invertible.
True. If A is not invertible, then Det [ A ] = 0 and Det [ B ] = 0
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35. If an nxn matrix A is invertible, then there must be an (n-1)x(n-1) submatrix of A (obtained by deleting a row and a column of A) that is invertible as well.
True. Det[ A ] = SUM (-1) i+j aij Det [ A ij].
Since Det[ A ] =/= 0, at least one of the
Det[ Aij ] must be non zero.
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36. If all the entries of matrices A and A -1 are integers, then the equation
Det (A) = Det (A -1 ) must hold.
True. Det [A] and Det[ A-1] are both integers
whose product is 1. They are both 1 or
both -1.
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37. If a square matrix A is invertible, then its classical adjoint adj(A) is invertible as well.
True. adj(A) =Det [A ] A -1 and its inverse is 1/Det[A] A.
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38. There is a 3x3 matrix A such that A 2 = -I3.
True | i 0 0 | Since A satisfies the
| 0 i 0 | polynomial is x 2 + 1 = 0
| 0 0 i | all the eigenvalues are
complex. A real matrix has to have one real root. Thus A cannot be real.
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39. There are invertible 3x3 matrices A and S
such that S -1 A S = 2 A.
False. Det [ S -1 A S ] = Det [A] =/= 2 n Det [A].
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40. There are invertible 3x3 matrices A and S
such that S T A S = -A
False. This would mean
Det [A] Det [S]2 = Det [-A] = - Det [A]
which is not possible when S is real.
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41. If all the diagonal entries of an nxn matrix A are odd integers and all the other entries are even integers, then A must be an invertible matrix.
True. In the determinant, there is only one
odd term and all the rest are even. Thus it
cannot be zero.
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42 If all the diagonal entries of an nxn matrix
A are even integers and all the other entries are odd integers, then A must be an invertible matrix.
False | -2 1 1 | | 1 | | 0 |
| 1 -2 1 | | 1 | = | 0 |
| 1 1 -2 | | 1 | | 0 |
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43. For every nonzero 2x2 matrix A there exists a 2x2 matrix B such that
Det[ A+B ]=/= Det[ A ]+Det [B ].
True. A = | a b | X = | x y |
| c d | | z w |
Det [A+X] – Det[A] – Det [X] = aw+dx-cy-bz
and if A =/= 0, we can make this nonzero.
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44. If A is a 4x4 matrix whose entries are all 1 or -1, then Det [A] must be divisible by 8.. (I.E. Det[A] = 8 k for some integer k.)
|1 1 1 1 | | a-1 b-1 c-1 |
True: Det |1 a b c | = Det| d-1 e-1 f-1 |
|1 d e f | | g-1 h-1 i-1 |
|1 g h i |
The entries in the 3x3 determinant are 0 or -2 and so a 2 can be factored out of each column.