Question
If $A =\left[\begin{array}{ll}3 & 7 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{ll}0 & 2 \\ 5 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}1 & -5 \\ -4 & 6\end{array}\right]$ Find $A B - 5C$
✓
Answer
Given : $A=\left[\begin{array}{ll}3 & 7 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{ll}0 & 2 \\ 5 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}1 & -5 \\ -4 & 6\end{array}\right]$
Now,
$\begin{array}{l} AB =\left[\begin{array}{ll}3 & 7 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}0 & 2 \\ 5 & 3\end{array}\right] \\ =\left[\begin{array}{ll}3 \times 0+7 \times 5 & 3 \times 2+7 \times 3 \\ 2 \times 0+4 \times 5 & 2 \times 2+4 \times 3\end{array}\right]\end{array}$
$\begin{array}{l}=\left[\begin{array}{ll}0+35 & 6+21 \\ 0+20 & 4+12\end{array}\right] \\ =\left[\begin{array}{ll}35 & 27 \\ 20 & 16\end{array}\right]\end{array}$
$5 C=5\left[\begin{array}{cc}1 & -5 \\ -4 & 6\end{array}\right]=\left[\begin{array}{cc}5 & -25 \\ -20 & 30\end{array}\right]$
$A B-5 C=\left[\begin{array}{ll}35 & 27 \\ 20 & 16\end{array}\right]-\left[\begin{array}{cc}5 & -25 \\ -20 & 30\end{array}\right]$
$=\left[\begin{array}{cc}30 & 52 \\ 40 & -14\end{array}\right]$
Now,
$\begin{array}{l} AB =\left[\begin{array}{ll}3 & 7 \\ 2 & 4\end{array}\right]\left[\begin{array}{ll}0 & 2 \\ 5 & 3\end{array}\right] \\ =\left[\begin{array}{ll}3 \times 0+7 \times 5 & 3 \times 2+7 \times 3 \\ 2 \times 0+4 \times 5 & 2 \times 2+4 \times 3\end{array}\right]\end{array}$
$\begin{array}{l}=\left[\begin{array}{ll}0+35 & 6+21 \\ 0+20 & 4+12\end{array}\right] \\ =\left[\begin{array}{ll}35 & 27 \\ 20 & 16\end{array}\right]\end{array}$
$5 C=5\left[\begin{array}{cc}1 & -5 \\ -4 & 6\end{array}\right]=\left[\begin{array}{cc}5 & -25 \\ -20 & 30\end{array}\right]$
$A B-5 C=\left[\begin{array}{ll}35 & 27 \\ 20 & 16\end{array}\right]-\left[\begin{array}{cc}5 & -25 \\ -20 & 30\end{array}\right]$
$=\left[\begin{array}{cc}30 & 52 \\ 40 & -14\end{array}\right]$
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