This document contains 44 true/false statements about determinants and properties of matrices. Some key points: - The determinant of a diagonal matrix is the product of its diagonal entries. - Interchanging two rows of a matrix changes the sign of the determinant. - Multiplying a column of a matrix by a constant c multiplies the determinant by c. - The determinant of a matrix is 0 if the matrix has identical rows or is not invertible. - For invertible matrices A and B, det(A) = det(B) if A and B are similar.

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 10. Det (A10) = (Det A) 10 for all 10x10
matrices A.
True. Det (AB) = Det (A) Det (B).

11. 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. 12. The determinant of all orthogonal
matrices is 1.
False. It is either 1 or -1.

13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 22. If A is any noninvertible square matrix,
then Det (A) = Det (rref(A).
True. Det (A) = 0. Det (rref(A)) = 0

23. 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. 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. 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. 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. 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. 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. 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. 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. 31, If A is any symmetric matrix, then
Det [A] = 1 or Det [A] = -1.
False Det | 0 2 | = -4
| 2 0 |

32. 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. +
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.