Mathematics 9 (Complete Solutions)

Q5
For the matrices, verify the following rules.

For the matrices       
A =
 
1 2 3
2 3 1
1 -1 0
 
,
B =
 
1 -1 1
2 -2 2
3 1 3
 
,
C =
 
-1 0 0
0 -2 3
1 1 2
 
verify the following rules.

(i) A + C = C + A
(ii) A + B = B + A
(iii) B + C = C + B
(iv) A + (B + A) = 2A + B
(v) (C - B) + A = C + (A - B)
(vi) 2A + B = A + (A + B)
(vii) (C - B) - A = (C - A) - B
(viii) (A + B) + C = A + (B + C)
(ix) A + (B - C) = (A - C) +B
(x) 2A + 2B = 2(A + B)

Solution:

(i) A + C = C + A
 
1 2 3
2 3 1
1 -1 0
 
+
 
-1 0 0
0 -2 3
1 1 2
 
=
 
-1 0 0
0 -2 3
1 1 2
 
+
 
1 2 3
2 3 1
1 -1 0
 
 
1-1 2+0 3+0
2+0 3-2 1+3
1+1 -1+1 0+2
 
=
 
-1+1 0+2 0+3
0+2 -2+3 3+1
1+1 1-1 2+0
 
 
0 2 3
2 1 4
2 0 2
 
=
 
0 2 3
2 1 4
2 0 2
 
Hence proved L.H.S = R.H.S

(ii) A + B = B + A
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
=
 
1 -1 1
2 -2 2
3 1 3
 
+
 
1 2 3
2 3 1
1 -1 0
 
 
1+1 2-1 3+1
2+2 3-2 1+2
1+3 -1+1 0+3
 
=
 
1+1 -1+2 1+3
2+2 -2+3 2+1
3+1 1-1 3+0
 
 
2 1 4
4 1 3
4 0 3
 
=
 
2 1 4
4 1 3
4 0 3
 
Hence proved L.H.S = R.H.S

(iii) B + C = C + B
 
1 -1 1
2 -2 2
3 1 3
 
+
 
-1 0 0
0 -2 3
1 1 2
 
=
 
-1 0 0
0 -2 3
1 1 2
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1-1 -1+0 1+0
2+0 -2-2 2+3
3+1 1+1 3+2
 
=
 
-1+1 0-1 0+1
0+2 -2-2 3+2
1+3 1+1 2+3
 
 
0 -1 1
2 -4 5
4 2 5
 
=
 
0 -1 1
2 -4 5
4 2 5
 
Hence proved L.H.S = R.H.S

(iv) A + (B + A) = 2A + B
 
1 2 3
2 3 1
1 -1 0
 
+
(
 
1 -1 1
2 -2 2
3 1 3
 
+
 
1 2 3
2 3 1
1 -1 0
 
)
=
2
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1 2 3
2 3 1
1 -1 0
 
+
 
1+1 -1+2 1+3
2+2 -2+3 2+1
3+1 1-1 3+0
 
=
 
2(1) 2(2) 2(3)
2(2) 2(3) 2(1)
2(1) 2(-1) 2(0)
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1 2 3
2 3 1
1 -1 0
 
+
 
2 1 4
4 1 3
4 0 3
 
=
 
2 4 6
4 6 2
2 -2 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1+2 2+1 3+4
2+4 3+1 1+3
1+4 -1+0 0+3
 
=
 
2+1 4-1 6+1
4+2 6-2 2+2
2+3 -2+1 0+3
 
 
3 3 7
6 4 4
5 -1 3
 
=
 
3 3 7
6 4 4
5 -1 3
 
Hence proved L.H.S = R.H.S

(v) (C - B) + A = C + (A - B)
(
 
-1 0 0
0 -2 3
1 1 2
 
-
 
1 -1 1
2 -2 2
3 1 3
 
)
+
 
1 2 3
2 3 1
1 -1 0
 
=
 
-1 0 0
0 -2 3
1 1 2
 
+
(
 
1 2 3
2 3 1
1 -1 0
 
-
 
1 -1 1
2 -2 2
3 1 3
 
)
 
-1-1 0+1 0-1
0-2 -2+2 3-2
1-3 1-1 2-3
 
+
 
1 2 3
2 3 1
1 -1 0
 
=
 
-1 0 0
0 -2 3
1 1 2
 
+
 
1-1 2+1 3-1
2-2 3+2 1-2
1-3 -1-1 0-3
 
 
-2 1 -1
-2 0 1
-2 0 -1
 
+
 
1 2 3
2 3 1
1 -1 0
 
=
 
-1 0 0
0 -2 3
1 1 2
 
+
 
0 3 2
0 5 -1
-2 -2 -3
 
 
-2+1 1+2 -1+3
-2+2 0+3 1+1
-2+1 0-1 -1+0
 
=
 
-1+0 0+3 0+2
0+0 -2+5 3-1
1-2 1-2 2-3
 
 
-1 3 2
0 3 2
-1 -1 -1
 
=
 
-1 3 2
0 3 2
-1 -1 -1
 
Hence proved L.H.S = R.H.S

(vi) 2A + B = A + (A + B)
2
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
(
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
)
 
2(1) 2(2) 2(3)
2(2) 2(3) 2(1)
2(1) 2(-1) 2(0)
 
+
 
1 -1 1
2 -2 2
3 1 3
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
 
1+1 2-1 3+1
2+2 3-2 1+2
1+3 -1+1 0+3
 
 
2 4 6
4 6 2
2 -2 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
 
2 1 4
4 1 3
4 0 3
 
 
2+1 4-1 6+1
4+2 6-2 2+2
2+3 -2+1 0+3
 
=
 
1+2 2+1 3+4
2+4 3+1 1+3
1+4 -1+0 0+3
 
 
3 3 7
6 4 4
5 -1 3
 
=
 
3 3 7
6 4 4
5 -1 3
 
Hence proved L.H.S = R.H.S

(vii) (C - B) - A = (C - A) - B
(
 
-1 0 0
0 -2 3
1 1 2
 
-
 
1 -1 1
2 -2 2
3 1 3
 
)
-
 
1 2 3
2 3 1
1 -1 0
 
=
(
 
-1 0 0
0 -2 3
1 1 2
 
-
 
1 2 3
2 3 1
1 -1 0
 
)
-
 
1 -1 1
2 -2 2
3 1 3
 
 
-1-1 0+1 0-1
0-2 -2+2 3-2
1-3 1-1 2-3
 
-
 
1 2 3
2 3 1
1 -1 0
 
=
 
-1-1 0-2 0-3
0-2 -2-3 3-1
1-1 1+1 2-0
 
-
 
1 -1 1
2 -2 2
3 1 3
 
 
-2 1 -1
-2 0 1
-2 0 -1
 
-
 
1 2 3
2 3 1
1 -1 0
 
=
 
-2 -2 -3
-2 -5 2
0 2 2
 
-
 
1 -1 1
2 -2 2
3 1 3
 
 
-2-1 1-2 -1-3
-2-2 0-3 1-1
-2-1 0+1 -1-0
 
=
 
-2-1 -2+1 -3-1
-2-2 -5+2 2-2
0-3 2-1 2-3
 
 
-3 -1 -4
-4 -3 0
-3 1 -1
 
=
 
-3 -1 -4
-4 -3 0
-3 1 -1
 
Hence proved L.H.S = R.H.S

(viii) (A + B) + C = A + (B + C)
(
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
)
+
 
-1 0 0
0 -2 3
1 1 2
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
(
 
1 -1 1
2 -2 2
3 1 3
 
+
 
-1 0 0
0 -2 3
1 1 2
 
)
 
1+1 2-1 3+1
2+2 3-2 1+2
1+3 -1+1 0+3
 
+
 
-1 0 0
0 -2 3
1 1 2
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
 
1-1 -1+0 1+0
2+0 -2-2 2+3
3+1 1+1 3+2
 
 
2 1 4
4 1 3
4 0 3
 
+
 
-1 0 0
0 -2 3
1 1 2
 
=
 
1 2 3
2 3 1
1 -1 0
 
+
 
0 -1 1
2 -4 5
4 2 5
 
 
2-1 1+0 4+0
4+0 1-2 3+3
4+1 0+1 3+2
 
=
 
1+0 2-1 3+1
2+2 3-4 1+5
1+4 -1+2 0+5
 
 
1 1 4
4 -1 6
5 1 5
 
=
 
1 1 4
4 -1 6
5 1 5
 
Hence proved L.H.S = R.H.S

(ix) A + (B - C) = (A - C) +B
 
1 2 3
2 3 1
1 -1 0
 
+
(
 
1 -1 1
2 -2 2
3 1 3
 
-
 
-1 0 0
0 -2 3
1 1 2
 
)
=
(
 
1 2 3
2 3 1
1 -1 0
 
-
 
-1 0 0
0 -2 3
1 1 2
 
)
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1 2 3
2 3 1
1 -1 0
 
+
 
1+1 -1-0 1-0
2-0 -2+2 2-3
3-1 1-1 3-2
 
=
 
1+1 2-0 3-0
2-0 3+2 1-3
1-1 -1-1 0-2
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1 2 3
2 3 1
1 -1 0
 
+
 
2 -1 1
2 0 -1
2 0 1
 
=
 
2 2 3
2 5 -2
0 -2 -2
 
+
 
1 -1 1
2 -2 2
3 1 3
 
 
1+2 2-1 3+1
2+2 3+0 1-1
1+2 -1+0 0+1
 
=
 
2+1 2-1 3+1
2+2 5-2 -2+2
0+3 -2+1 -2+3
 
 
3 1 4
4 3 0
3 -1 1
 
=
 
3 1 4
4 3 0
3 -1 1
 
Hence proved L.H.S = R.H.S

(x) 2A + 2B = 2(A + B)
2
 
1 2 3
2 3 1
1 -1 0
 
+
2
 
1 -1 1
2 -2 2
3 1 3
 
=
2 (
 
1 2 3
2 3 1
1 -1 0
 
+
 
1 -1 1
2 -2 2
3 1 3
 
)
 
2(1) 2(2) 2(3)
2(2) 2(3) 2(1)
2(1) 2(-1) 2(0)
 
+
 
2(1) 2(-1) 2(1)
2(2) 2(-2) 2(2)
2(3) 2(1) 2(3)
 
=
2
 
1+1 2-1 3+1
2+2 3-2 1+2
1+3 -1+1 0+3
 
 
2 4 6
4 6 2
2 -2 0
 
+
 
2 -2 2
4 -4 4
6 2 6
 
=
2
 
2 1 4
4 1 3
4 0 3
 
 
2+2 4-2 6+2
4+4 6-4 2+4
2+6 -2+2 0+6
 
=
 
2(2) 2(1) 2(4)
2(4) 2(1) 2(3)
2(4) 2(0) 2(3)
 
 
4 2 8
8 2 6
8 0 6
 
=
 
4 2 8
8 2 6
8 0 6
 

Hence proved L.H.S = R.H.S

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