Question

If $\left[\begin{array}{cc}a+2b& 2c-d\\ c+4d& 4b-a\end{array}\right]=\left[\begin{array}{cc}10& 2\\ 37& 14\end{array}\right]$, then find the values of $a,b,c,d$ respectively.

Answer 1

$\left[\begin{array}{cc}a+2b& 2c-d\\ c+4d& 4b-a\end{array}\right]=\left[\begin{array}{cc}10& 2\\ 37& 14\end{array}\right]$

By the equality of two matrices, equating the corresponding elements, we get
$a+2b=10...\left(1\right)2c-d=2...\left(2\right)c+4d=37...\left(3\right)4b-a=14...\left(4\right)$.

Adding eqn(1) and (4), we get
$6b=24\Rightarrow b=4\Rightarrow a=2$.

Multiplying eqn(2) by $4$ and add eqn(3), we get
$8c-4d+c+4d=45\Rightarrow 9c=45\Rightarrow c=5\Rightarrow d=8$.

The values are $a=2,b=4,c=5,d=8$.