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

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