Question

If $A=\left[\begin{array}{cc}2& 7\\ 1& 5\end{array}\right]$ then what is $A+3A^{-1}$ equal to?
where $I$ is the identity matrix of order $2$.

• A. $3I$
• B. $5I$
• C. $7I$
• D. None of the above

Answer 1

The correct option is C $7I$

Inverse of a matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

$\Rightarrow$ Inverse of $\left[\begin{array}{cc}2& 7\\ 1& 5\end{array}\right]=\frac{1}{10-7}\left[\begin{array}{cc}5& -7\\ -1& 2\end{array}\right]$

$\Rightarrow A^{-1}=\frac{1}{3}\left[\begin{array}{cc}5& -7\\ -1& 2\end{array}\right]$

$A+3A^{-1}=\left[\begin{array}{cc}2& 7\\ 1& 5\end{array}\right]+\left[\begin{array}{cc}5& -7\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right]=7I$