Solveeit Logo
Login

Question

Mathematics Question on Matrices

If A=[−123\579−211]\begin{bmatrix}-1&2&3\\\5&7&9\\\\-2&1&1\end{bmatrix}and B=[−41−5\120\131]\begin{bmatrix}-4&1&-5\\\1&2&0\\\1&3&1\end{bmatrix},then verify that
(i)(A+B)'=A'+B'
(ii)(A-B)'=A'-B'

Answer

We have: A'=[−15−2\271\391]\begin{bmatrix}-1&5&-2\\\2&7&1\\\3&9&1\end{bmatrix},B'=[−411\123−501]\begin{bmatrix}-4&1&1\\\1&2&3\\\\-5&0&1\end{bmatrix}

(i)A+B= [−123\579−211]\begin{bmatrix}-1&2&3\\\5&7&9\\\\-2&1&1\end{bmatrix}+[−41−5\120\131]\begin{bmatrix}-4&1&-5\\\1&2&0\\\1&3&1\end{bmatrix}

=[−53−2\699\142]\begin{bmatrix}-5&3&-2\\\6&9&9\\\1&4&2\end{bmatrix}

therefore (A+B)'=[−56−1\394−292]\begin{bmatrix}-5&6&-1\\\3&9&4\\\\-2&9&2\end{bmatrix}

A'+B'=[−15−2\271\391]\begin{bmatrix}-1&5&-2\\\2&7&1\\\3&9&1\end{bmatrix}+[−411\123−501]\begin{bmatrix}-4&1&1\\\1&2&3\\\\-5&0&1\end{bmatrix}
Hence we verified that (A+B)'=A'+B'
(ii)A-B=[−123\579−211]\begin{bmatrix}-1&2&3\\\5&7&9\\\\-2&1&1\end{bmatrix}-[−41−5\120\131]\begin{bmatrix}-4&1&-5\\\1&2&0\\\1&3&1\end{bmatrix}

=[318\459−3−20]\begin{bmatrix}3&1&8\\\4&5&9\\\\-3&-2&0\end{bmatrix}

therefore (A-B)'=[−34−3\15−2\890]\begin{bmatrix}-3&4&-3\\\1&5&-2\\\8&9&0\end{bmatrix}

A'-B'=[−15−2\271\391]\begin{bmatrix}-1&5&-2\\\2&7&1\\\3&9&1\end{bmatrix}-[−411\123−501]\begin{bmatrix}-4&1&1\\\1&2&3\\\\-5&0&1\end{bmatrix}

=[−34−3\15−2\890]\begin{bmatrix}-3&4&-3\\\1&5&-2\\\8&9&0\end{bmatrix}

Hence we verified that (A-B)'=A'-B'