Matrix Quiz
Put T if the statement is always true, otherwise put F .
O is the zero matrix and I is the identity matrx.
All variables are assumed to be well-defined unless otherwise stated. All equations are linear.
There is no need to prove if the statement is true. Thin of a !ounter-example if it is false.
"""""1. If A is a square matrix and A
#
$ I, then A $ I or A $ -I.
"""""2. If A% $ O, then A $ O or % $ O.
"""""3. If A, %, & are square and A%& $ O, then one of them is O .
"""""4. If A% $ A&, then % $ &.
"""""5. If A is non-zero and A% $ A&, then % $ &.
"""""6. The square of a non-zero square matrix must be a non-zero matrix
"""""7. If A% $ %A, then 'A t %(
)
$ A
)
t )A
#
% * )A%
#
t %
)
.
"""""8. An invertible matrix must be square.
"""""9. 'A%(
-+
$ A
-+
%
-+
.
"""""10. If A has a zero row or zero !olumn, then A is not invertible.
"""""11. If A is a square matrix whi!h has no zero rows or zero !olumns, then A is invertible.
"""""12. ,et A, % be invertible matri!es of same size. Then A% is also invertible.
"""""13. ,et A, % be invertible matri!es of same size. Then A t % is also invertible.
"""""14. If A% is equal to the identity matrix, then A must be invertible matrix.
"""""15. If A, % are square matri!es. If A% $ I, then %A $ I. -en!e, A is invertible.
"""""16. .or square matrix A, AA
T
$ I if and only if A
T
A $ I
"""""17. If A% is invertible, then %A is invertible.
"""""18. If A
#
O , then A is invertible.
"""""19. If A is invertible, then A
#
O .
"""""20. If A is a square matrix and A
#
* /A 0 I $ O , then A is invertible.
"""""21. A symmetri! matrix must be a square matrix.
"""""22. If A is symmetri!, so are A
-+
'if exists( and A
)
.
+
"""""23. If % $ A
T
A, then )% is symmetri!.
"""""24. If A is symmetri!, so is p'A( , for any polynomial p'x(.
"""""25. det 'A t %( $ det A t det %.
"""""26. det 'A( $ det A.
"""""27. Three elementary row operations do not !han1e the determinant of a square matrix.
"""""28. If A is row equivalent to %, then det A $ det %.
"""""29. ,et A is row equivalent to %, then det A and det % are either both zero or both nonzero.
"""""30. ,et A be a square matrix without zero rows and !olumns. Then A must be row equivalent to the
identity matrix of the same size.
"""""31. If A, % are nonzero square matri!es and A is row equivalent to % , then both A, % are invertible.
"""""32. If A is an invertible matrix and % is row equivalent to A, then % is also invertible.
"""""33. 'A t %(
T
$ A
T
t %
T
.
"""""34. 'A %(
T
$ A
T
%
T
.
"""""35. If m 2 n, then the system A
mn
x $ O always has a nontrivial solution.
"""""36. If A is an m n matrix with m 3 n, then Ax $ O always has a nontrivial solution.
"""""37. If det A 4, then the system of equation Ax $ O has non-trivial solution.
"""""38. ,et A be an n n matrix. If the equation has a unique solution for a 1iven nonzero
n + ve!tor b , then det A 4.
"""""39. ,et A be m n matrix. If the equation Ax $ b has a unique solution for a nonzero ve!tor m +,
then the homo1eneous equation Ax $ O has only trivial solution.
"""""40. If det A $ 4, then the system of equation Ax $ O has no solution.
"""""41. If A , % are square matri!es and A is not invertible, then A% is not invertible.
"""""42. If row # of a determinant is repla!ed by row + 0 row # , the determinant remains the same.
"""""43. In a system of ) linear equations and ) variables x, y, z , if the determinants
$ x
$ y
$ z
$ 4, then the systems of equations has infinite numbers of solutions.
"""""44.
a b !
! a b
b ! a
x y z
z x y
y z x
!an be expressed in the form A%& ) & % A
) ) )
+ + , where A, %, & are
#
fun!tions of x, y, z, a, b and !.
Solution
1. F , A $
,
_
4 +
+ 4
, A
#
$
,
_
,
_
,
_
+ 4
4 +
4 +
+ 4
4 +
+ 4
$ I, A I and A - I.
2. F ,
,
_
,
_
,
_
4 4
4 4
A% ,
+ 4
4 4
4 4
% ,
4 4 4
4 4 +
A
55 A $
,
_
4 4
4 +
, % $
,
_
+ 4
4 4
, A% $
,
_
4 4
4 4
3. F , A $ % $ & $
,
_
4 +
4 4
, A%& $ O
4. F , &hoose A $ O, % &
5. F , A $
,
_
4 4
4 +
, % $
,
_
4 4
4 4
, & $
,
_
+ 4
4 4
, A% $ A& $
,
_
4 4
4 4
, % &.
6. F ,
,
_
,
_
4 4
4 4
4 4
+ 4
#
7. T
8. T , A is invertible A% $ %A $ I . If A is m n and % is n m, then m $ n.
9. F , 'A%(
-+
$ %
-+
A
-+
sin!e 'A%(' %
-+
A
-+
( $ I.
10. T , det A $ 4 and hen!e not invertible.
11. F ,
,
_
+ +
+ +
has no zero row or zero !olumn and has no inverse.
12. T , A% is defined and square. det 'A%( $ det A det % 4 .
13. F , A $
,
_
+ 4
4 +
% $
,
_
+ 4
4 +
, A * % $
,
_
4 4
4 4
14. F , A $
,
_
+ 4 4
4 4 +
% $
,
_
+ 4
4 4
4 +
, A% $
,
_
+ 4
4 +
$ I
15. T
16. T
17. F , A $
,
_
+ 4 4
4 4 +
% $
,
_
+ 4
4 4
4 +
, A% $
,
_
+ 4
4 +
$ I , %A $
,
_
+ 4 4
4 4 4
4 4 +
I
18. F , A $
,
_
# +
# +
, A
#
O , det A $ 4 A
-+
does not exist.
19. T
20. T , A is square and A'A * /I( $ I A
-+
$ A * /I
)
21. T
22. T 'A
-+
(
T
$ 'A
T
(
-+
, 'A
n
(
T
$ 'A
T
(
n
.
23. T
24. T
25. F , A $
,
_
+ 4
4 +
% $
,
_
+ 4
4 +
det A $ +, det % $ +, det 'A * %( $ 4
26. F , A $
,
_
+ 4
4 +
det A $ +, det 'A( $ det
,
_
4
4
$
#
27. F , +. If % is obtained from A by inter!han1in1 two rows of A, then det % $ - det A
#. If % is obtained from A by multiplyin1 a row in A by , then det % $ det A
). If % is obtained from A by !han1in1 6
i
by 6
i
* 6
7
, det % $ det A .
28. F , A $
,
_
8 )
# +
, then A 9
,
_
+ 4
4 +
$ I , det A $ -# + .
29. T
30. F , A $
,
_
+ +
+ +
, then A 9
,
_
4 4
+ +
31. F , A $
,
_
+ +
+ +
, then A 9
,
_
4 4
+ +
$ %
32. T
33. T
34. F , 'A%(
T
$ %
T
A
T
.
35. T
36. F ,
,
_
,
_
,
_
4
4
y
x
4 4
+ 4
4 +
has only trivial solution .
37. F , :hould be det A $ 4 .
38. T
39. T
40. F ,
'
+
+
# y # x #
+ y x
has infinite number of solutions.
41. T
42. F , the new determinant is the ne1ative of the ori1inal determinant.
43. F ,
'
+ +
+ +
+ +
+ z 4 y 4 x 4
4 z 4 y 4 x 4
4 z 4 y 4 x 4
has no solution.
8
44. T , A ax * bz * !y , % ay * bx * !z, & az * by * !x .
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