Question

Given $A=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right].$ Find all possible matrix $X$ for which $AX=A.$

• A. $X=\left[\begin{array}{cc}a& b\\ 2-2a& 1-2b\end{array}\right]$ for a, $b\in R$
• B. $X=\left[\begin{array}{cc}b& a\\ 2-2b& 1-2a\end{array}\right]$ for a, $b\in R$
• C. $X=\left[\begin{array}{cc}b& a\\ 1-2a& 2-2b\end{array}\right]$ for a, $b\in R$
• D. none of these

Answer 1

Answer: (A), (B)

The correct options are
A $X=\left[\begin{array}{cc}b& a\\ 2-2b& 1-2a\end{array}\right]$ for a, $b\in R$
B $X=\left[\begin{array}{cc}a& b\\ 2-2a& 1-2b\end{array}\right]$ for a, $b\in R$

Given $A=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]$
We will check by options
Option A:
Consider, $AX=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ 2-2a& 1-2b\end{array}\right]$
$=\left[\begin{array}{cc}2a+2-2a& 2b+1-2b\\ 2a+2-2a& 2b+1-2b\end{array}\right]$

$\Rightarrow AX=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]$

$\Rightarrow AX=A$
Hence, this matrix X satisfies given condition.
Option B:
Consider $AX=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]\left[\begin{array}{cc}b& a\\ 2-2b& 1-2a\end{array}\right]$

$=\left[\begin{array}{cc}2b+2-2b& 2a+1-2a\\ 2b+2-2b& 2a+1-2a\end{array}\right]$

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

$\Rightarrow AX=A$
Hence, this matrix X also satisfies the given equation.
Option C:
Consider, $AX=\left[\begin{array}{cc}2& 1\\ 2& 1\end{array}\right]\left[\begin{array}{cc}b& a\\ 1-2a& 2-2b\end{array}\right]$
$=\left[\begin{array}{cc}2b+1-2a& 2a+2-2b\\ 2b+1-2a& 2a+2-2b\end{array}\right]$
$\Rightarrow AX\ne A$
Hence, option C is not possible matrix X.