Question

Given matrix $A=\left[\begin{array}{cc}4\sin {30}^{\circ }& \cos 0^{\circ }\\ \cos 0^{\circ }& 4\sin {30}^{\circ }\end{array}\right]$ and $B=\left[\begin{array}{c}4\\ 5\end{array}\right]$. If $AX=B$.
Find the matrix $X$.

Answer 1

Let $X=\left[\begin{array}{c}a\\ b\end{array}\right]$

Therefore, $\left[\begin{array}{cc}4\sin {30}^o& \cos 0^o\\ \cos 0^o& 4\sin {30}^o\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}4\\ 5\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}4a\sin {30}^o+b\cos 0^o\\ a\cos 0^o+4b\sin {30}^o\end{array}\right]=\left[\begin{array}{c}4\\ 5\end{array}\right]$

$\Rightarrow 4a\sin {30}^{\circ }+b\cos 0^{\circ }=4\Rightarrow 4a\times \frac{1}{2}+b\times 1=4$

$\Rightarrow 2a+b=4$ ... (1)
$\Rightarrow a\cos 0^{\circ }+4b\sin {30}^{\circ }=5\Rightarrow 9\times 1+4b\times \frac{1}{2}=5$
$\Rightarrow a+2b=5$ ..... (2)
Solving eqs (1) and (2), we get
$a=1$ and $b=2$
Hence, $X=\left[\begin{array}{c}1\\ 2\end{array}\right]$