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Mathematics Question on Matrices

Let A= [24\32]\begin {bmatrix} 2&4\\\3&2\end {bmatrix},B=[13−25]\begin {bmatrix} 1&3\\\\-2&5\end {bmatrix},C=[−25\34]\begin {bmatrix} -2&5\\\3&4\end {bmatrix}.Find each of the following

I. A+B
II. A-B
III. 3A-C
IV. AB
V. BA

Answer

(i) A+B=[24\32]\begin {bmatrix} 2&4\\\3&2\end {bmatrix}+[13−25]\begin {bmatrix} 1&3\\\\-2&5\end {bmatrix}=[2+14+3\3−22+5]\begin{bmatrix}2+1&4+3\\\3-2&2+5\end{bmatrix}=[37\17]\begin {bmatrix} 3&7\\\1&7\end {bmatrix}

(ii) A-B= [24\32]\begin {bmatrix} 2&4\\\3&2\end {bmatrix}-[13−25]\begin {bmatrix} 1&3\\\\-2&5\end {bmatrix}=[2−14−3\3−(−2)2−5]\begin{bmatrix}2-1&4-3\\\3-(-2)&2-5\end{bmatrix}=[11\5−3]\begin {bmatrix} 1&1\\\5&-3\end {bmatrix}

(iii) 3A-C=3[24\32]\begin {bmatrix} 2&4\\\3&2\end {bmatrix}-[−25\34]\begin {bmatrix} -2&5\\\3&4\end {bmatrix}

=[3∗23∗4\3∗33∗2]\begin{bmatrix}3*2&3*4\\\3*3&3*2\end{bmatrix}-[−25\34]\begin {bmatrix} -2&5\\\3&4\end {bmatrix}

=[612\96]\begin{bmatrix}6&12\\\9&6\end{bmatrix}-[−25\34]\begin {bmatrix} -2&5\\\3&4\end {bmatrix}=[6+212−5\9−36−4]\begin{bmatrix}6+2&12-5\\\9-3&6-4\end{bmatrix}=[87\62]\begin {bmatrix} 8&7\\\6&2\end {bmatrix}

(iv)Matrix A has 2 columns. This number is equal to the number of rows in matrix B.
Therefore, AB is defined as:
AB=\begin {bmatrix} 2&4\\\3&2\end {bmatrix}$$\begin {bmatrix} 1&3\\\\-2&5\end {bmatrix}

=[2(1)+4(−2)2(3)+4(5)\3(1)+2(−2)3(3)+2(5)]\begin{bmatrix}2(1)+4(-2)&2(3)+4(5)\\\3(1)+2(-2)&3(3)+2(5)\end{bmatrix}

=[2−86+20\3−49+10]\begin{bmatrix}2-8&6+20\\\3-4&9+10\end{bmatrix}=[−626−119]\begin {bmatrix} -6&26\\\\-1&19\end {bmatrix}

(v) Matrix B has 2 columns. This number is equal to the number of rows in matrix A.
Therefore, BA is defined as:

BA=\begin {bmatrix} 1&3\\\\-2&5\end {bmatrix}$$\begin {bmatrix} 2&4\\\3&2\end {bmatrix}

=[1(2)+3(3)1(4)+3(2)−2(2)+5(3)−2(4)+5(2)]\begin{bmatrix}1(2)+3(3)&1(4)+3(2)\\\\-2(2)+5(3)&-2(4)+5(2)\end{bmatrix}

=[2+94+6−4+15−8+10]=[1110\112]\begin{bmatrix}2+9&4+6\\\\-4+15&-8+10\end{bmatrix}=\begin{bmatrix}11&10\\\11&2\end{bmatrix}