Question

Given matrix $B=\left[\begin{array}{cc}1& 1\\ 8& 3\end{array}\right]$. Find the matrix X if, $X=B^2-4B$. Hence solve for a and b given $X\left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}5\\ 50\end{array}\right]$.

Answer 1

$B=$ $\left[\begin{array}{cc}1& 1\\ 8& 3\end{array}\right]$

and it is given

$X$ $=$ $B^2$ $-4B$

$B^2$ $=$ $\left[\begin{array}{cc}1& 1\\ 8& 3\end{array}\right]$$\left[\begin{array}{cc}1& 1\\ 8& 3\end{array}\right]$

$B^2$ $=$ $\left[\begin{array}{cc}1+8& 1+3\\ 8+24& 8+9\end{array}\right]$=$\left[\begin{array}{cc}9& 4\\ 32& 17\end{array}\right]$

$4B$ $=$ $4$ $\left[\begin{array}{cc}1& 1\\ 8& 3\end{array}\right]$=$\left[\begin{array}{cc}4& 4\\ 32& 12\end{array}\right]$

$B^2$$-4B=$ $\left[\begin{array}{cc}9& 4\\ 32& 17\end{array}\right]$-$\left[\begin{array}{cc}4& 4\\ 32& 12\end{array}\right]$=$\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

And further$\left[\begin{array}{c}a\\ b\end{array}\right]$ $=$ $\left[\begin{array}{c}5\\ 50\end{array}\right]$

$\Rightarrow$ $\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$ $\left[\begin{array}{c}a\\ b\end{array}\right]$ $=$ $\left[\begin{array}{c}5\\ 50\end{array}\right]$

$\Rightarrow$ $\left[\begin{array}{c}5a+0\\ 0+5b\end{array}\right]$ $=$ $\left[\begin{array}{c}5\\ 50\end{array}\right]$

$\Rightarrow$ $\left[\begin{array}{c}5a\\ 5b\end{array}\right]$ $=$ $\left[\begin{array}{c}5\\ 50\end{array}\right]$

hence $5a=5$ so, $a=1$ and $5b=50$ so, $b=10$