Question

Let $X=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$, $A=\left[\begin{array}{ccc}1& -1& 2\\ 2& 0& 1\\ 3& 2& 1\end{array}\right]$ and $B=\left[\begin{array}{c}3\\ 1\\ 4\end{array}\right]$. If $AX=B$, then X is equal to.

• A. $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$
• B. $\left[\begin{array}{c}-1\\ -2\\ 3\end{array}\right]$
• C. $\left[\begin{array}{c}-1\\ -2\\ -3\end{array}\right]$
• D. $\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$

Answer 1

Answer: (D)

$\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$

Since, $A=\left[\begin{array}{ccc}1& -1& 2\\ 2& 0& 1\\ 3& 2& 1\end{array}\right]$
Now, $|A|=1(0-2)+(2-3)+2(4-0)$
$=-2-1+8=5$
$\therefore A^{-1}=\frac{1}{5}\left[\begin{array}{ccc}-2& 5& -1\\ 1& -5& 3\\ 4& -5& 2\end{array}\right]$
Now, $A^{-1}B=\frac{1}{5}\left[\begin{array}{ccc}-2& 5& -1\\ 1& -5& 3\\ 4& -5& 2\end{array}\right]$ $\left[\begin{array}{c}3\\ 1\\ 4\end{array}\right]$
$\Rightarrow \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$