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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