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# Let $A\;=\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}, B\;=\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}, C\;=\begin{bmatrix}-2 & 5\\3 & 4\end{bmatrix}$$Find\;each\;of\;the\;following:$$(i)\;A+B\qquad(ii)\;(A-B)\qquad(iii)\;3A-C$$(iv)\;AB\qquad(v)BA$

(i)Since A & B are of the same order $2\times 2.$Hence addition of A&B is defined and given by

A+B$\Rightarrow \begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}+\begin{bmatrix}2+1 & 4+3\\3-2 & 2+5\end{bmatrix}.$

$\;\;\;=\begin{bmatrix}2+1 & 4+3\\3-2 & 2+5\end{bmatrix}\Rightarrow\begin{bmatrix}3 & 7\\1 & 7\end{bmatrix}.$

(ii)A-B

If A=$[a_{ij}],B=[b_{ij}]$ are two matrices of the same order say $m\times n$ then difference A-B is defined by D.

D=A-B$\Rightarrow A+(-1)B.$

i.e sum of the matrix A and the matrix B

A-B=A+(-B)

Given:A=$\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}B=\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}.$

A-B=$\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}-1\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}.$

Therefore A+(-B)=$\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}+\begin{bmatrix}-1 & -3\\2 &- 5\end{bmatrix}.$

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

(iii)3A-C

3A=$3\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}\Rightarrow \begin{bmatrix}2\times 3 & 4\times 3\\3\times 3 & 3\times 2\end{bmatrix}$

$\Rightarrow \begin{bmatrix}2 & -5\\-3 & -4\end{bmatrix}$

Since A&C are of the same order $2\times 2$

Therefore Addition of 3A&-C is given by

$3A-C=\begin{bmatrix}6 & 12\\9 & 6\end{bmatrix}+\begin{bmatrix}2 &-5\\-3 &-4\end{bmatrix}$

$\Rightarrow \begin{bmatrix}6+2 & 12-5\\9-3 & 6-4\end{bmatrix}\Rightarrow\begin{bmatrix}8 & 7\\6 & 2\end{bmatrix}$

(iv)AB

The product of two matrices A&B is defined if the number of columns of A is equal to that of the number of rows of B.

$\Rightarrow Given A=\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}B=\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}$

the above given value of A & B .The number of column of A is equal to the number of rows of B.So the product of the matrices A & B is defined as

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

$\Rightarrow\begin{bmatrix}2\times 1+4\times -2& 2\times 3+4\times 5\\3\times 1+2\times -2 & 3\times 3+2\times 5\end{bmatrix}$

$\Rightarrow \begin{bmatrix}2-3& 6+20\\3-4 & 9+10\end{bmatrix}\Rightarrow \begin{bmatrix}-6& 26\\-1 & 19\end{bmatrix}$

(v)BA

Here no of column of B is equal to the no of row of B.So the product of matrices B and A is defined

$\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}\begin{bmatrix}2& 4\\3 & 2\end{bmatrix}\Rightarrow \begin{bmatrix}1\times 2+3\times 3 & 1\times 4+3\times 2\\-2\times 2+5\times 3 & -2\times 4+5\times 2\end{bmatrix}$

$\Rightarrow \begin{bmatrix}2+9& 4+6\\-4+15 & -8+10\end{bmatrix}\Rightarrow \begin{bmatrix}11& 10\\11 & 2\end{bmatrix}$