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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