Question
If $A=\left[\begin{array}{ccc}7 & 3 & 0 \\ 0 & 4 & -2\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 3 \\ 2 & 1 & -4\end{array}\right]$, then find
i. $A^{\top}+4 B^{\top}$
ii. $5 A^{\top}-5 B^{\top}$
i. $A^{\top}+4 B^{\top}$
ii. $5 A^{\top}-5 B^{\top}$
$\therefore \quad A^T=\left[\begin{array}{cc}7 & 0 \\ 3 & 4 \\ 0 & -2\end{array}\right]$ and $B^T=\left[\begin{array}{cc}0 & 2 \\ -2 & 1 \\ 3 & -4\end{array}\right]$
$A^T+4 B^T=\left[\begin{array}{cc}7 & 0 \\ 3 & 4 \\ 0 & -2\end{array}\right]+4\left[\begin{array}{cc}0 & 2 \\ -2 & 1 \\ 3 & -4\end{array}\right]$
$=\left[\begin{array}{cc}7 & 0 \\ 3 & 4 \\ 0 & -2\end{array}\right]+\left[\begin{array}{cc}0 & 8 \\ -8 & 4 \\ 12 & -16\end{array}\right]$
$=\left[\begin{array}{cc}7+0 & 0+8 \\ 3-8 & 4+4 \\ 0+12 & -2-16\end{array}\right]=\left[\begin{array}{cc}7 & 8 \\ -5 & 8 \\ 12 & -18\end{array}\right]$
$5 A^{\top}-5 B^{\top}=5\left(A^{\top}-B^{\top}\right)$
$=5\left(\left[\begin{array}{cc}7 & 0 \\ 3 & 4 \\ 0 & -2\end{array}\right]-\left[\begin{array}{cc}0 & 2 \\ -2 & 1 \\ 3 & -4\end{array}\right]\right)$
$=5\left[\begin{array}{cc}7-0 & 0-2 \\ 3+2 & 4-1 \\ 0-3 & -2+4\end{array}\right]=5\left[\begin{array}{cc}7 & -2 \\ 5 & 3 \\ -3 & 2\end{array}\right]=\left[\begin{array}{cc}35 & -10 \\ 25 & 15 \\ -15 & 10\end{array}\right]$
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