Prove that

Question:

Let $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$. Find:

i. $A+2 B$

ii. $B-4 c$

iii. $A-2 B+3 C$

Solution:

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

$=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]+\left[\begin{array}{cc}2 & 6 \\ -4 & 10\end{array}\right]$

$=\left[\begin{array}{cc}4 & 10 \\ -1 & 12\end{array}\right]$

Conclusion: $(\mathrm{A}+2 \mathrm{~B})=\left[\begin{array}{cc}4 & 10 \\ -1 & 12\end{array}\right]$

ii. $B-4 c$

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

$=\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]-\left[\begin{array}{cc}-8 & 20 \\ 12 & 16\end{array}\right]$

$=\left[\begin{array}{cc}9 & -17 \\ -14 & -11\end{array}\right]$

Conclusion: $\mathrm{A}_{-} 2 \mathrm{~B}+3 \mathrm{C}=\left[\begin{array}{cc}-6 & 13 \\ 16 & 4\end{array}\right]$