Question

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

Answer 1

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

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

(iii) $3A-C=3\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$

$=\left[\begin{array}{cc}3\times 2& 3\times 4\\ 3\times 3& 3\times 2\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$

$=\left[\begin{array}{cc}6& 12\\ 9& 6\end{array}\right]-\left[\begin{array}{cc}-2& 5\\ 3& 4\end{array}\right]$

$=\left[\begin{array}{cc}6+2& 12-5\\ 9-3& 6-4\end{array}\right]$

$=\left[\begin{array}{cc}8& 7\\ 6& 2\end{array}\right]$

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

$AB=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]=\left[\begin{array}{cc}2\left(1\right)+4(-2)& 2\left(3\right)+4\left(5\right)\\ 3\left(1\right)+2(-2)& 3\left(3\right)+2\left(5\right)\end{array}\right]$

$=\left[\begin{array}{cc}2-8& 6+20\\ 3-4& 9+10\end{array}\right]\left[\begin{array}{cc}-6& 26\\ -1& 19\end{array}\right]$

(v) Matrix $B$ has $2$ columns, This number is equal to the number of rows in matrix $A$. Therefore, $BA$ is defined as :
$BA=\left[\begin{array}{cc}1& 3\\ -2& 5\end{array}\right]\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]=\left[\begin{array}{cc}1\left(2\right)+3\left(3\right)& 1\left(4\right)3\left(2\right)\\ -2\left(2\right)+5\left(3\right)& -2\left(4\right)+5\left(2\right)\end{array}\right]$

$=\left[\begin{array}{cc}2+9& 4+6\\ -4+15& -8+10\end{array}\right]=\left[\begin{array}{cc}11& 10\\ 11& 2\end{array}\right]$