Question

Which of the following matrices is NOT left inverse of matrix $\left[\begin{array}{cc}1& -1\\ 1& 1\\ 2& 3\end{array}\right]$?

• A. $\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$
• B. $\left[\begin{array}{ccc}2& -7& 3\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$
• C. $\left[\begin{array}{ccc}-\frac{1}{2}& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$
• D. $\left[\begin{array}{ccc}0& 3& -1\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{ccc}0& 3& -1\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]$

We will find $LA$ for each option.
Given, $A=\left[\begin{array}{cc}1& -1\\ 1& 1\\ 2& 3\end{array}\right]$
For option A,
$LA=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 1\\ 2& 3\end{array}\right]$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow LA=I$
So, option A is left inverse of A.
For option B
$LA=\left[\begin{array}{ccc}2& -7& 3\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 1\\ 2& 3\end{array}\right]$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow LA=I$
Hence, option B is the left inverse of A.
For option C,
$LA=\left[\begin{array}{ccc}-\frac{1}{2}& \frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}& 0\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 1\\ 2& 3\end{array}\right]$
$=\left[\begin{array}{cc}0& 1\\ 0& 1\end{array}\right]$
$\Rightarrow LA\ne I$
Hence, option C is not the left inverse of A.
Option D,
$\Rightarrow LA=I$
Hence, option D is the left inverse of A.