If A=⎡⎢⎣−123579−211⎤⎥⎦ and B=⎡⎢⎣−41−5120131⎤⎥⎦, then verify that:
(i) (A+B)′=A′+B′
(ii) (A−B)′=A′−B′
(i) (A+B)′=A′+B′
(ii) (A−B)′=A′−B′
(i) Given: A=⎡⎢⎣−123579−211⎤⎥⎦ and B=⎡⎢⎣−41−5120131⎤⎥⎦
To verify: (A+B)′=A′+B′
Solving L.H.S.
(A+B)′=⎛⎜⎝⎡⎢⎣−123579−211⎤⎥⎦+⎡⎢⎣−41−5120131⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎛⎜⎝⎡⎢⎣−1+(−4)2+13+(−5)5+17+29+0−2+11+31+1⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎛⎜⎝⎡⎢⎣−53−2699−142⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎡⎢⎣−56−1394−292⎤⎥⎦
Solving R.H.S.
A′+B′=⎡⎢⎣−123579−211⎤⎥⎦′+⎡⎢⎣−41−5120131⎤⎥⎦′
A′+B′=⎡⎢⎣−15−2271391⎤⎥⎦+⎡⎢⎣−411123−501⎤⎥⎦
⇒A′+B′=⎡⎢⎣−1+(−4)5+1−2+12+17+21+33+(−5)9+01+1⎤⎥⎦
⇒A′+B′=⎡⎢⎣−56−1394−292⎤⎥⎦
So, L.H.S.=R.H.S.
Hence verified.
(ii) Given: A=⎡⎢⎣−123579−211⎤⎥⎦ and B=⎡⎢⎣−41−5120131⎤⎥⎦
To verify: (A−B)′=A′−B′
Solving L.H.S.
(A−B)′=⎛⎜⎝⎡⎢⎣−123579−211⎤⎥⎦−⎡⎢⎣−41−5120131⎤⎥⎦⎞⎟⎠′
⇒(A−B)′=⎛⎜⎝⎡⎢⎣−1−(−4)2−13−(−5)5−17−29−0−2−11−31−1⎤⎥⎦⎞⎟⎠
⇒(A−B)′=⎛⎜⎝⎡⎢⎣318459−3−20⎤⎥⎦⎞⎟⎠′
⇒(A−B)′=⎡⎢⎣34−315−2890⎤⎥⎦
Solving R.H.S.
A′−B′=⎡⎢⎣−123579−211⎤⎥⎦′−⎡⎢⎣−41−5120131⎤⎥⎦′
⇒A′−B′=⎡⎢⎣−1−(−4)5−1−2−12−17−21−33−(−5)9−01−1⎤⎥⎦
⇒A′−B′=⎡⎢⎣34−315−2890⎤⎥⎦
So, L.H.S.=R.H.S.
Hence verified.
(i) Given: A=⎡⎢⎣−123579−211⎤⎥⎦ and B=⎡⎢⎣−41−5120131⎤⎥⎦
To verify: (A+B)′=A′+B′
Solving L.H.S.
(A+B)′=⎛⎜⎝⎡⎢⎣−123579−211⎤⎥⎦+⎡⎢⎣−41−5120131⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎛⎜⎝⎡⎢⎣−1+(−4)2+13+(−5)5+17+29+0−2+11+31+1⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎛⎜⎝⎡⎢⎣−53−2699−142⎤⎥⎦⎞⎟⎠′
⇒(A+B)′=⎡⎢⎣−56−1394−292⎤⎥⎦
Solving R.H.S.
A′+B′=⎡⎢⎣−123579−211⎤⎥⎦′+⎡⎢⎣−41−5120131⎤⎥⎦′
A′+B′=⎡⎢⎣−15−2271391⎤⎥⎦+⎡⎢⎣−411123−501⎤⎥⎦
⇒A′+B′=⎡⎢⎣−1+(−4)5+1−2+12+17+21+33+(−5)9+01+1⎤⎥⎦
⇒A′+B′=⎡⎢⎣−56−1394−292⎤⎥⎦
So, L.H.S.=R.H.S.
Hence verified.
(ii) Given: A=⎡⎢⎣−123579−211⎤⎥⎦ and B=⎡⎢⎣−41−5120131⎤⎥⎦
To verify: (A−B)′=A′−B′
Solving L.H.S.
(A−B)′=⎛⎜⎝⎡⎢⎣−123579−211⎤⎥⎦−⎡⎢⎣−41−5120131⎤⎥⎦⎞⎟⎠′
⇒(A−B)′=⎛⎜⎝⎡⎢⎣−1−(−4)2−13−(−5)5−17−29−0−2−11−31−1⎤⎥⎦⎞⎟⎠
⇒(A−B)′=⎛⎜⎝⎡⎢⎣318459−3−20⎤⎥⎦⎞⎟⎠′
⇒(A−B)′=⎡⎢⎣34−315−2890⎤⎥⎦
Solving R.H.S.
A′−B′=⎡⎢⎣−123579−211⎤⎥⎦′−⎡⎢⎣−41−5120131⎤⎥⎦′
⇒A′−B′=⎡⎢⎣−1−(−4)5−1−2−12−17−21−33−(−5)9−01−1⎤⎥⎦
⇒A′−B′=⎡⎢⎣34−315−2890⎤⎥⎦
So, L.H.S.=R.H.S.
Hence verified.
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