Mathematics 9 (Complete Solutions)

# Q5 verify whether

Let
A =

 -1 3 2 0

,
B =

 1 2 -3 -5

and
C =

 2 1 1 3

verify whether.

 (i) AB = BA (ii) A(BC) = (AB)C (iii) A(B + C) = AB + AC (iv) A(B - C) = AB - AC

Solution:

(i) AB = BA

 -1 3 2 0

 1 2 -3 -5

=

 1 2 -3 -5

 -1 3 2 0

 -1(1) + 3(-3) -1(2) + 3(-5) 2(1) + 0(-3) 2(2) + 0(-5)

=

 1(-1) + (2)(2) 1(3) + (2)(0) -3(-1) + (-5)(2) -3(3) + (-5)(0)

 -1 - 9 -2 - 15 2 + 0 4 + 0

=

 -1 + 4 3 + 0 3 - 10 -9 + 0

 -10 -17 2 4

=

 3 3 -7 -9

Hence verified that AB ≠ BA

(ii) A(BC) = (AB)C

 -1 3 2 0

(

 1 2 -3 -5

 2 1 1 3

) = (

 -1 3 2 0

 1 2 -3 -5

)

 2 1 1 3

 -1 3 2 0

 1(2) + 2(1) 1(1) + 2(3) -3(2) + (-5)(1) -3(1) + (-5)(3)

=

 -1(1) + 3(-3) -1(2) + 3(-5) 2(1) + 0(-3) 2(2) + 0(-5)

 2 1 1 3

 -1 3 2 0

 2 + 2 1 + 6 -6 - 5 -3 - 15

=

 -1 - 9 -2 - 15 2 + 0 4 + 0

 2 1 1 3

 -1 3 2 0

 4 7 -11 -18

=

 -10 -17 2 4

 2 1 1 3

 -1(4) + 3(-11) -1(7) + 3(-18) 2(4) + 0(-11) 2(7) + 0(-18)

=

 -10(2) + (-17)(1) -10(1) + (-17)(3) 2(2) + 4(1) 2(1) + 4(3)

 -4 - 33 -7 - 54 8 + 0 14 + 0

=

 -20 + -17 -10 - 51 4 + 4 2 + 12

 -37 -61 8 14

=

 -37 -61 8 14

Hence verified that A(BC) = (AB)C

(iii) A(B + C) = AB + AC

 -1 3 2 0

(

 1 2 -3 -5

+

 2 1 1 3

) =

 -1 3 2 0

 1 2 -3 -5

+

 -1 3 2 0

 2 1 1 3

 -1 3 2 0

 1+2 2+1 -3+1 -5+3

=

 -1(1) + 3(-3) -1(2) + 3(-5) 2(1) + 0(-3) 2(2) + 0(-5)

+

 -1(2) + 3(1) -1(1) + 3(3) 2(2) + 0(1) 2(1) + 0(3)

 -1 3 2 0

 3 3 -2 -2

=

 -1 - 9 -2 - 15 2 + 0 4 + 0

+

 -2 + 3 -1 + 9 4 + 0 2 + 0

 -1(3) + 3(-2) -1(3) + 3(-2) 2(3) + 0(-2) 2(3) + 0(-2)

=

 -10 -17 2 4

+

 1 8 4 2

 -3 - 6 -3 - 6 6 + 0 6 + 0

=

 -10+1 -17+8 2+4 4+2

 -9 -9 6 6

=

 -9 -9 6 6

Hence verified that A(B + C) = AB + AC

(iv) A(B - C) = AB - AC

 -1 3 2 0

(

 1 2 -3 -5

-

 2 1 1 3

) =

 -1 3 2 0

 1 2 -3 -5

-

 -1 3 2 0

 2 1 1 3

 -1 3 2 0

 1-2 2-1 -3-1 -5-3

=

 -1(1) + 3(-3) -1(2) + 3(-5) 2(1) + 0(-3) 2(2) + 0(-5)

+

 -1(2) + 3(1) -1(1) + 3(3) 2(2) + 0(1) 2(1) + 0(3)

 -1 3 2 0

 -1 1 -4 -8

=

 -1 - 9 -2 - 15 2 + 0 4 + 0

-

 -2 + 3 -1 + 9 4 + 0 2 + 0

 -1(-1) + 3(-4) -1(1) + 3(-8) 2(-1) + 0(-4) 2(1) + 0(-8)

=

 -10 -17 2 4

+

 1 8 4 2

 1 - 12 -1 - 24 -2 + 0 2 + 0

=

 -10-1 -17-8 2-4 4-2

 -11 -25 -2 2

=

 -11 -25 -2 2

Hence verified that A(B - C) = AB - AC