# if

Question:

If $A=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right], B=\left[\begin{array}{rrr}-1 & 0 & 2 \\ 3 & 4 & 1\end{array}\right], C=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 2 & 1 & 0\end{array}\right]$, find

(i) $A+B$ and $B+C$

(ii) $2 B+3 A$ and $3 C-4 B$.

Solution:

(i)

$A+B=\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right]+\left[\begin{array}{ccc}-1 & 0 & 2 \\ 3 & 4 & 1\end{array}\right]$

It is not possible to add these matrices because the number of elements in A are not equal to the
number of elements in B. So, A + B does not exist.

$\Rightarrow B+C=\left[\begin{array}{ccc}-1 & 0 & 2 \\ 3 & 4 & 1\end{array}\right]+\left[\begin{array}{ccc}-1 & 2 & 3 \\ 2 & 1 & 0\end{array}\right]$

$\Rightarrow B+C=\left[\begin{array}{cc}-1-1 & 0+2 & 2+3 \\ 3+2 & 4+1 & 1+0\end{array}\right]$

$\Rightarrow B+C=\left[\begin{array}{ccc}-2 & 2 & 5 \\ 5 & 5 & 1\end{array}\right]$

(ii)

$2 B+3 A=2\left[\begin{array}{ccc}-1 & 0 & 2 \\ 3 & 4 & 1\end{array}\right]+3\left[\begin{array}{ll}2 & 3 \\ 5 & 7\end{array}\right]$

It is not possible to add these matrices because the number of elements in B are not equal to the
number of elements in A. So, 2B + 3A does not exist.

$\Rightarrow 3 C-4 B=3\left[\begin{array}{ccc}-1 & 2 & 3 \\ 2 & 1 & 0\end{array}\right]-4\left[\begin{array}{ccc}-1 & 0 & 2 \\ 3 & 4 & 1\end{array}\right]$

$\Rightarrow 3 C-4 B=\left[\begin{array}{ccc}-3 & 6 & 9 \\ 6 & 3 & 0\end{array}\right]-\left[\begin{array}{ccc}-4 & 0 & 8 \\ 12 & 16 & 4\end{array}\right]$

$\Rightarrow 3 C-4 B=\left[\begin{array}{ccc}-3+4 & 6-0 & 9-8 \\ 6-12 & 3-16 & 0-4\end{array}\right]$

$\Rightarrow 3 C-4 B=\left[\begin{array}{ccc}1 & 6 & 1 \\ -6 & -13 & -4\end{array}\right]$