Question

If $A=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$, then $\left(\text{adj}A\right)A=$

• A. $\left[\begin{array}{cc}\frac{1}{5}& 0\\ 0& \frac{1}{5}\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}5& 0\\ 0& -5\end{array}\right]$
• D. $\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

$A=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$

$\therefore \text{adj}A=\left[\begin{array}{cc}(-1{)}^{1+1}4& (-1{)}^{1+2}1\\ (-1{)}^{1+2}3& (-1{)}^{2+2}2\end{array}\right]$

$\therefore \text{adj}A=\left[\begin{array}{cc}4& -1\\ -3& 2\end{array}\right]$

$\therefore \left(\text{adj}A\right)A=\left[\begin{array}{cc}4& -1\\ -3& 2\end{array}\right]\times \left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$

$=\left[\begin{array}{cc}4\times 2-1\times 3& 4\times 1-1\times 4\\ -3\times 2+2\times 3& -3\times 1+2\times 4\end{array}\right]$

$=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$