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SUMMER BREAK ASSIGNMENT (2012-13)MATHEMATICS
TOPIC: MATRICES
Let A =
21
32and f(x) = 742 + xx , then show that f(A) = O , using this
result findA5 and A-1.
If A =
14
3
, B = [ ]543 , prove that (AB)T = BTAT.
If A and B are symmetric matrices of same order , show thatAB +
BA is a symmetric matrix.AB BA is a skew -symmetric matrix.
Let A =
00
10, then by P.M.I. prove that ( ) bAnaIabAaI nnn 1+=+
If A =
24
53
, verify that A2
5A 14 I = O, and hence find A-1
.
Show that A =
211
121
112
satisfies the equation A3 -6A2 +9A 4 I = O.
7. Using row transformation , find the inverse of following
matrices .
i.
74
32ii.
121
232
405
8 Using elementary transformation find the inverse of
223
221
111
matrix and hence solve
x + y + z = 4: x 2y +2z = -1 ; 3x + 2y 2z = 5.
9. Solve the following system of homogeneous equations:i. 3 x 4y
+ 5z = 0; x + y 2z = 0 ; 3x + 3y + z = 0ii. x + y z = 0; x 2y + z =
0; 3x + 6y 5z = 0
10. Find the value of for which the following system of
equations has non-trivial solutions. Also,
find the solution .2x + 3y 2z = 0; 2x y + 3z = 0; 7x + y z =
011. Check the consistency of following system of equations, if
consistent then find the solution also.
i. x + y + z = 6; x + 2y + 3z = 14; x + 4y + 7z = 30ii. 3x y 2z
= 2; 2y z = -1 ; 3x 5y = 3iii x - y = 3; 2x + 3y + 4z = 17 ; y + 2z
= 7
TOPIC : DETERMINANTS
1. Using properties of determinants show that
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i. ))()((2 accbba
cbaab
acbac
bccba
+++=++
++++
ii.222
2222
2222
2222
4 cba
acbb
acba
ccba
=+
++
iii. Prove that
=
2sin
2sin
2sin4
sincos1
sincos1
sincos1BAACCB
CC
BB
AA
iv. Find 0 2 so that 0
4sin41cossin
4sin4cos1sin
4sin4cossin1
22
22
22
=
+
++
v If A + B + C = , then )sin()sin()sin(coscossinsin
coscossinsin
coscossinsin
22
22
22
ACCBBA
CCCC
BBBB
AAAA
=
vi Without expanding the determinant at any stage show that
,
121232
333132
21
2
2
2
BxA
xxxx
xxxx
xxxx
+=+++++
where A and b are determinants of order 3 not
involving x.
vii If p + q + r = 0, then prove that
acb
bac
cba
pqr
qapcrb
pbraqc
rcqbpa
=
viii Prove that 0
0
0
0
=
lmpq
mlab
qpba
ix Prove that for all , 0
32sin
3
2cos
3
2sin
3
4sin
3
2cos
3
2sin
2sincossin
=
+
+
+
RELATION AND FUNCTIONS
1. Let A = N N and R be a relation on A defined as (a, b) R (c,
d) ad = bc. Show that R is anequivalence relation.
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2. Let N be set of natural numbers and R be a relation on N N
defined as(a, b) R (c, d) ad(b + c) = bc ( a + d). Show that R is
an equivalence relation.
3. Show that f : N N given by f(x) =
+1
1
x
x
if
ifx
xis
iseven
oddis a bijective function.
4. A = R {3} , B = R {1}. Show that f : A B defined by f(x)
=3
2
x
xis bijective function. Also
define 1f .
5. Consider f : ),5[ +R given by f(x) = 569 2 + xx . Show that f
is invertible also find f-1.
6. Show that f(x) =
+
,3
,1
x
x
32
20
b > c > 0, then prove that
3. Prove that
4. Solve for x:
i.
ii.
iii. 2
