Question

Assertion :Let $A=\left[\begin{array}{ccc}0& c& -b\\ -c& 0& a\\ b& -a& 0\end{array}\right]$ and $t\in C$.

$adj(tI-A)=t^{2}I+tA+A^2+(a^2+b^2+c^2)I$

Reason: $adjA=A^2+(a^2+b^2+c^2)I$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$A=\left[\begin{array}{ccc}0& c& -b\\ -c& 0& a\\ b& -a& 0\end{array}\right]$
$\Rightarrow (tI-A)=\left[\begin{array}{ccc}t& -c& b\\ c& t& -a\\ -b& a& t\end{array}\right]$
$adj(tI-A)=\left[\begin{array}{ccc}{t}^2+a^2& tc+ab& ac-tb\\ tc-ab& t^2+b^2& at+bc\\ ac+tb& -at+bc& t^2+c^2\end{array}\right]$
$=t^2\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+t\left[\begin{array}{ccc}0& c& -b\\ -c& 0& a\\ b& -a& 0\end{array}\right]+\left[\begin{array}{ccc}-c^2-b^2& ab& ac\\ ab& -c^2-a^2& bc\\ ac& bc& -b^2-a^2\end{array}\right]+(a^2+b^2+c^2)\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\therefore adj(tI-A)=t^{2}I+tA+A^2+(a^2+b^2+c^2)I$
Assertion is true.
$adjA=\left[\begin{array}{ccc}{a}^2& ab& ac\\ ab& b^2& bc\\ ac& bc& c^2\end{array}\right]$
$=\left[\begin{array}{ccc}-c^2-b^2& ab& ac\\ ab& -c^2-a^2& bc\\ ac& bc& -b^2-a^2\end{array}\right]+(a^2+b^2+c^2)\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\therefore adjA=A^2+(a^2+b^2+c^2)I$
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.