Question

Let $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right],C=\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right],$ then find
$\left(i\right)$ $A+B$
$\left(ii\right)$ $A-B$
$\left(iii\right)$ $3A-C$
$\left(iv\right)$ $AB$
$\left(v\right)$ $BA$

Answer 1

$\left(i\right):$ Given: $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
Now, $A+B=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]+\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
$=\left[\begin{array}{cc}2+1& 4+3\\ 3-2& 2+5\end{array}\right]$
$=\left[\begin{array}{cc}3& 7\\ 1& 7\end{array}\right]$

$\left(ii\right)$ Given: $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
Now, $A-B=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]-\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
$=\left[\begin{array}{cc}2-1& 4-3\\ 3-(-2)& 2-5\end{array}\right]$
$=\left[\begin{array}{cc}1& 1\\ 3+2& -3\end{array}\right]$
$=\left[\begin{array}{cc}1& 1\\ 5& -3\end{array}\right]$

$\left(iii\right)$ Given: $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],C=\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$
Now, $3A-C=3\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$
$=\left[\begin{array}{cc}3\times 2& 3\times 4\\ 3\times 3& 3\times 2\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$
$=\left[\begin{array}{cc}6& 12\\ 9& 6\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$
$=\left[\begin{array}{cc}6+2& 12-5\\ 9-3& 6-4\end{array}\right]$
$=\left[\begin{array}{cc}8& 7\\ 6& 2\end{array}\right]$

$\left(iv\right)$ Given: $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
Now $AB=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]=\left[\begin{array}{cc}1& 3\\ -1& 5\end{array}\right]$
$=\left[\begin{array}{cc}2\times 1+4\times -2& 2\times 3+4\times 5\\ 3\times 1+2\times -2& 3\times 3+2\times 5\end{array}\right]$
$=\left[\begin{array}{cc}2-8& 6+20\\ 3-4& 9+10\end{array}\right]$
$=\left[\begin{array}{cc}-6& 26\\ -1& 19\end{array}\right]$

$\left(v\right)$ Given: $A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right],B=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]$
Now, $BA=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]$
$=\left[\begin{array}{cc}1\times 2+3\times 3& 1\times 4+3\times 2\\ -2\times 2+5\times 3& -2\times 4+5\times 2\end{array}\right]$
$=\left[\begin{array}{cc}2+9& 4+6\\ -4+15& -8+10\end{array}\right]$
$=\left[\begin{array}{cc}11& 10\\ 11& 2\end{array}\right]$