1.Algebra Booster

7.6 Algebra Booster
where
Proof:
fi
fi
fi
a1 b1 c1 d1 b1 c1
D = a2 b2 c2 , D1=
d2 b2 c2
,
a3 b3 c3 d3 b3 c3
a1 d1 c1 a1 b1 d1
D2= a2 d2 c2 and D3=
a2 b2 d2
a3 d3 c3 a3 b3 d3
a1 b1 c1
Given D = a2 b2 c2
a3 b3 c3
a1 b1 c1
xD = x a2 b2 c2
a3 b3 c3
ax 1 b1 c1
= ax 2 b2 c2
ax 3 b3 c3
ax 1 + by 1 + cz 1 b1 c1
= ax 2 + by 2 + cz 2 b2 c2
ax 3 + by 3 + cz 3 b3 c3
(C 1
Æ C 1
+ C 2
+ C 3
)
d1 b1 c1
xD = d2 b2 c2 = D1
d b c
3 3 3
D1
fi x =
D
Similarly, we can proved that
D 2 D3
y = , z =
D D
Hence, the result.
Nature of solutions of the system of equaions by Camers Rule
(i) If D π 0, the system of equations has a unique solution
and is said to be consistent.
(ii) If D = 0 as well as D 1
= 0 = D 2
= D 3
, the system of
equations has infinitely many solutions and is said to
be consistent.
(iii) If D = 0 and at least one of D 1
, D 2
, D 3
is non-zero, the
system of equations has no solution and is said to be
inconsistent.
8. HOMOGENEOUS SYSTEM OF EQUATIONS
The given homogenous system of equations are
a 1
x + b 1
y + c 1
z = 0
a 2
x + b 2
y + c 2
z = 0
a 3
x + b 3
y + c 3
z = 0.
a1 b1 c1 0 b1 c1
Let D = a2 b2 c2 , D1=
0 b2 c2
,
a b c 0 b c
3 3 3 3 3
a1 0 c1 a1 b1
0
D2 = a2 0 c2 and D3=
a2 b2
0
a 0 c a b 0
3 3 3 3
Nature of solutions by homogeneous system of equations
(i) If D π 0, the system of equations has only trivial solution,
say x = 0 = y = z, and the system of equations is
said to be consistent.
(ii) If D = 0, the system of equations has non-trivial solution,
i.e. infinite solutions and the system of equations
is also said to be consistent.
9. MULTIPLICATION OF TWO DETERMINANTS
Two determinants can be multiplied by a variety of ways.
row-by-column, row-by-row, column-by-column and column-by-row
multiplication rule.
1 1 1 1
Let A= a b and B =
m n .
a b m n
Then
2 2 2 2
a b m n
AB = ¥ a b m n
1 1 1 1
2 2 2 2
10. DIFFERENTIATION OF DETERMINANT
Let
Then
or
f() x g() x h()
x
Fx () = px () qx () rx ()
ux () vx () wx ()
f ¢ () x g¢ () x h¢
() x
F¢
() x = px () qx () rx ()
ux () vx () wx ()
f() x g() x h()
x
+ p¢ () x q¢ () x r¢
() x
ux () vx () wx ()
f() x g() x h()
x
+ px () qx () rx ()
u¢ () x v¢ () x w¢
() x
f ¢ ( x) g( x) h( x)
F¢ ( x) = p¢
( x) q( x) r( x)
u¢
( x) v( x) w( x)
f( x) g¢
( x) h( x)
+ px ( ) q¢
( x) rx ( )
ux ( ) v¢
( x) wx ( )
f( x) g( x) h¢
( x)
+ px ( ) qx ( ) r¢
( x)
ux ( ) vx ( ) w¢
( x)