Question

Assertion :If a, b, c, d are real numbers and $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $A^3=O$, then $A^2=O$. Reason: For matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, we have $A^2-(a+d)A+(ad-bc)I=O$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]A.T.P{A}^2-(a+d)A+(ad-bc)I=0\Rightarrow \left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]-(a+d)\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+(ad-bc)\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\Rightarrow \left[\begin{array}{cc}{a}^2+bc-a^2-ad+ad-bc& ab+bd-ab-bd\\ ac+cd-ac-cd& bc+d^2-ad-d^2+ad-bc\end{array}\right]\Rightarrow \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\therefore$ option $2$ is correct

Now $A^3=0\therefore {\left|A\right|}^3=0\Rightarrow \left|A\right|=0\therefore \left|A\right|=0\Rightarrow (ad-bc)=0$

Now as per question $A^2-(a+d)A+(ad-bc)I=0$

$\therefore A^2-(a+d)A=0\Rightarrow A^2=(a+d)A$

So if $a+d=0$ then $A^2=0$

if $a+d\ne 0$,

(We know $A^3=0$)

So $A^{2}A$

$(a+d)AA=0$

$(a+d)A^2=0$ (but $a+d\ne 0$)

$\therefore A^2=0$