Evaluate the following

Question:

Evaluate the following

(i) $\left(\left[\begin{array}{rr}1 & 3 \\ -1 & -4\end{array}\right]+\left[\begin{array}{rr}3 & -2 \\ -1 & 1\end{array}\right]\right)\left[\begin{array}{lll}1 & 3 & 5 \\ 2 & 4 & 6\end{array}\right]$

(ii) $\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]\left[\begin{array}{l}2 \\ 4 \\ 6\end{array}\right]$

(iii) $\left[\begin{array}{rr}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]\left(\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 0 & 1\end{array}\right]-\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 0 & 2\end{array}\right]\right)$

Solution:

(i) $\left(\left[\begin{array}{cc}1 & 3 \\ -1 & -4\end{array}\right]+\left[\begin{array}{cc}3 & -2 \\ -1 & 1\end{array}\right]\right)\left[\begin{array}{lll}1 & 3 & 5 \\ 2 & 4 & 6\end{array}\right]$

$\Rightarrow\left(\left[\begin{array}{cc}1+3 & 3-2 \\ -1-1 & -4+1\end{array}\right]\right)\left[\begin{array}{lll}1 & 3 & 5 \\ 2 & 4 & 6\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}4 & 1 \\ -2 & -3\end{array}\right]\left[\begin{array}{ccc}1 & 3 & 5 \\ 2 & 4 & 6\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}4+2 & 12+4 & 20+6 \\ -2-6 & -6-12 & -10-18\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}6 & 16 & 26 \\ -8 & -18 & -28\end{array}\right]$

(ii) $\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]\left[\begin{array}{l}2 \\ 4 \\ 6\end{array}\right]$

$\Rightarrow\left[\begin{array}{lll}1+4+0 & 0+0+3 & 2+2+6\end{array}\right]\left[\begin{array}{l}2 \\ 4 \\ 6\end{array}\right]$

$\Rightarrow\left[\begin{array}{lll}5 & 3 & 10\end{array}\right]\left[\begin{array}{l}2 \\ 4 \\ 6\end{array}\right]$

$\Rightarrow[10+12+60]$

$\Rightarrow[82]$

(iii) $\left[\begin{array}{cc}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]\left(\left[\begin{array}{lll}1 & 0 & 2 \\ 2 & 0 & 1\end{array}\right]-\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 0 & 2\end{array}\right]\right)$

$\Rightarrow\left[\begin{array}{cc}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{ccc}1-0 & 0-1 & 2-2 \\ 2-1 & 0-0 & 1-2\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}1 & -1 \\ 0 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{ccc}1 & -1 & 0 \\ 1 & 0 & -1\end{array}\right]$

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

$\Rightarrow\left[\begin{array}{ccc}0 & -1 & 1 \\ 2 & 0 & -2 \\ 5 & -2 & -3\end{array}\right]$

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