Question
If $$A$$ is any $$2 \times 2$$ matrix such that \[\left[ {\begin{array}{*{20}{c}}
1&2\\
0&3
\end{array}} \right]A = \left[ {\begin{array}{*{20}{c}}
{ - 1}&0\\
6&3
\end{array}} \right],\] then what is $$A$$ equal to ?
A.
\[\left[ {\begin{array}{*{20}{c}}
{ - 5}&1\\
{ - 2}&2
\end{array}} \right]\]
B.
\[\left[ {\begin{array}{*{20}{c}}
{ - 5}&{ - 2}\\
{ 1}&2
\end{array}} \right]\]
C.
\[\left[ {\begin{array}{*{20}{c}}
{ - 5}&{ - 2}\\
{ 2}&1
\end{array}} \right]\]
D.
\[\left[ {\begin{array}{*{20}{c}}
{5}&2\\
{ - 2}&{ - 1}
\end{array}} \right]\]
Answer :
\[\left[ {\begin{array}{*{20}{c}}
{ - 5}&{ - 2}\\
{ 2}&1
\end{array}} \right]\]
Solution :
\[\begin{array}{l}
{\rm{Let,}}\,\,\,\left[ {\begin{array}{*{20}{c}}
1&2\\
0&3
\end{array}} \right] = B\\
{\rm{Then,}}\,\,BA = \left[ {\begin{array}{*{20}{c}}
{ - 1}&0\\
6&3
\end{array}} \right]\\
\Rightarrow \,A = {B^{ - 1}}\left[ {\begin{array}{*{20}{c}}
{ - 1}&0\\
{ - 6}&3
\end{array}} \right]\left| B \right| = 3,\\
adj\,B = \left[ {\begin{array}{*{20}{c}}
3&{ - 2}\\
0&1
\end{array}} \right]\\
{B^{ - 1}} = \frac{1}{3}\left[ {\begin{array}{*{20}{c}}
3&{ - 2}\\
0&1
\end{array}} \right]\\
\Rightarrow \,A = \frac{1}{3}\left[ {\begin{array}{*{20}{c}}
3&{ - 2}\\
0&3
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{ - 1}&0\\
6&3
\end{array}} \right] = \frac{1}{3}\left[ {\begin{array}{*{20}{c}}
{ - 3 - 12}&{ - 6}\\
6&3
\end{array}} \right]\\
= \,\left[ {\begin{array}{*{20}{c}}
{ - 5}&{ - 2}\\
2&1
\end{array}} \right]
\end{array}\]