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.

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

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.

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.

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.

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.

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.

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.

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.

10. Det (A10) = (Det A) 10 for all 10x10 matrices A.

True. Det (AB) = Det (A) Det (B).

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).

12. The determinant of all orthogonal matrices is 1.

False. It is either 1 or -1.

13. If A is any n x n matrix, then

Det( A A T) = Det( A T A )

True. Both equal Det(A) 2

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.

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.

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.

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

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 )

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.

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.

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.

22. If A is any noninvertible square matrix, then Det (A) = Det (rref(A).

True. Det (A) = 0. Det (rref(A)) = 0

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 |

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 |

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 |

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 |

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.

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 |

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 |

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)

31, If A is any symmetric matrix, then

Det [A] = 1 or Det [A] = -1.

False Det | 0 2 | = -4

| 2 0 |

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.

+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 |

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

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.

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.

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.

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.

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].

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.

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.

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 |

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.

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.