Question

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $A^2-(a+d)A-(bc-ad)I=$

• A. $0$
• B. $I$
• C. $2I$
• D. $(a-d)$

Answer 1

The correct option is A $0$

$A-\lambda I=\left[\begin{array}{cc}a-I\lambda & b\\ c& d-\lambda \end{array}\right]$
$|A-\lambda I|=|a-\lambda |(d-\lambda )-bc$
$=ad-\lambda (a+d)+{\lambda }^2-bc$
$={\lambda }^2-\lambda (a+d)-(bc-ad)$
Putting A in place of $\lambda$ we get
$0=A^2-A(a+d)-(bc-ad)$