Mathematics 9 (Complete Solutions)

Q6
For the matrices

A =
 
-1 3
2 0
 
,
B =
 
1 2
-3 -5
 
,
C =
 
-2 6
3 -9
 
verify that.

(i) (AB)t = BtAt
(ii) (BC)t = CtBt

Solution:


(i) (AB)t = BtAt
(
 
-1 3
2 0
 
 
1 2
-3 -5
 
)t
=
(
 
1 2
-3 -5
 
)t
(
 
-1 3
2 0
 
)t
(
 
-1(1) + 3(-3) -1(2) + 3(-5)
2(1) + 0(-3) 2(2) + 0(-5)
 
)t
=
 
1 -3
2 -5
 
 
-1 2
3 0
 
(
 
-1 - 9 -2 - 15
2 + 0 4 + 0
 
)t
=
 
1(-1) + (-3)(3) 1(2) + (-3)(0)
2(-1) + (-5)(3) 2(2) + (-5)(0)
 
(
 
-10 -17
2 4
 
)t
=
 
-1 - 9 2 + 0
-2 - 15 4 + 0
 
 
-10 2
-17 4
 
=
 
-10 2
-17 4
 
Hence verified that (AB)t = BtAt


(ii) (BC)t = CtBt
(
 
1 2
-3 -5
 
 
-2 6
3 -9
 
)t
=
(
 
-2 6
3 -9
 
)t
(
 
1 2
-3 -5
 
)t
(
 
1(-2) + 2(3) 1(6) + 2(-9)
-3(-2) + (-5)(3) -3(6) + (-5)(-9)
 
)t
=
 
-2 3
6 -9
 
 
1 -3
2 -5
 
(
 
-2 + 6 6 - 18
6 - 15 -18 + 45
 
)t
=
 
1(-2) + (2)(3) -3(-2) + (-5)(3)
1(6) + 2(-9) -3(6) + (-5)(-9)
 
(
 
4 -12
-9 27
 
)t
=
 
-2 + 6 6 - 15
6 - 18 -18 + 45
 
 
4 -9
-12 27
 
=
 
4 -9
-12 27
 
Hence verified that (BC)t = CtBt

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