Question

The number of right inverses for the matrix $\left[\begin{array}{ccc}1& -1& 2\\ 2& -1& 1\end{array}\right]$ is

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

The correct option is D infinite

$A$ is a matrix of order m$\times$n. If there exists a matrix $R$ of type n $\times$ m such that $AR=I_m$, then $R$ is called right inverse of $A$.
Let $R=\left[\begin{array}{cc}a& d\\ b& e\\ c& f\end{array}\right]$
$\left[\begin{array}{ccc}1& -1& 2\\ 2& -1& 1\end{array}\right]\left[\begin{array}{cc}a& d\\ b& e\\ c& f\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$a-b+2c=1,d-e+2f=0;2a-b+c=0,2d-e+f=1$
Three variables in two equations, hence above system has infinite solutions.
$\therefore$ Infinite right inverses are possible.
Hence, option D.