Question

Assertion :As $A=\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 1\\ 1& 1& 2\end{array}\right]$ satisfies the equation $x^3-5x^2+7x-3=0$, therefore $A$ is invertible. Reason: If a square matrix $A$ satisfies the equation $a_0{x}^n+a_1{x}^{n-1}+...a_{n-1}x+a_n=0$, and $a_n\ne 0$, then $A$ is invertible.

• 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

Since
$a_0{A}^n+a_1{A}^{n-1}+...+a_{n-1}A+a_{n}I=0$, and $a_n\ne 0$,
we get $AB=I$ where
$B=-\frac{a_0}{a_n}{A}^{n-1}-\frac{a_1}{a_n}{A}^{n-2}-...-\frac{a_{n-1}}{a_n}I$
$\Rightarrow B=A^{-1}$.
Given $A$ satisfies $A^3-5A^2+7A-3I=0$
$\Rightarrow A(A^2-5A+7I)=3I$
$\Rightarrow B=\frac{1}{3}(A^2-5A+7I)$
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.