Question

If $P=\left[\begin{array}{ccc}2& 3& 1\\ 0& -1& 5\end{array}\right]$ and $Q=\left[\begin{array}{ccc}1& 2& -6\\ 0& -1& 3\end{array}\right]$, Evaluate $3P-4Q$, if
$3P-4Q=\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]$, find sum of $a,b,c,d,e,f$

Answer 1

Since, $P=\left[\begin{array}{ccc}2& 3& 1\\ 0& -1& 5\end{array}\right]$ and $Q=\left[\begin{array}{ccc}1& 2& -6\\ 0& -1& 3\end{array}\right]$

$\therefore 3P-4Q=3\left[\begin{array}{ccc}2& 3& 1\\ 0& -1& 5\end{array}\right]-4\left[\begin{array}{ccc}1& 2& -6\\ 0& -1& 3\end{array}\right]$

$=\left[\begin{array}{ccc}6& 9& 3\\ 0& -3& 15\end{array}\right]-\left[\begin{array}{ccc}4& 8& -24\\ 0& -4& 12\end{array}\right]$

$=\left[\begin{array}{ccc}6-4& 9-8& 3+24\\ 0-0& -3+4& 15-12\end{array}\right]$

$=\left[\begin{array}{ccc}2& 1& 27\\ 0& 1& 3\end{array}\right]$

Now, $3P-4Q=\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}2& 1& 27\\ 0& 1& 3\end{array}\right]=\left[\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right]$

On comparing both sides, we get

$\Rightarrow a=2,b=1,c=27$

$\Rightarrow d=0,e=1,f=3$

Hence, the answer is $a=2,b=1,c=27,d=0,e=1,f=3.$ and their sum is $34$