Question

Assertion :If $A=\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$ then $adj\left(adjA\right)=A$ Reason: $|adj\cdot (adj\cdot A\left)\right|=$ ${\left|A\right|}^{(n-1{)}^2},A$ is a non singular matrix of order n

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R)is false,
• D. (A)is false but (R) is true.

Answer 1

The correct option is B Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

$A=\left[\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right]$
$adj\left(adjA\right)=|A{|}^{(n-2)}A$
$|A|=\left|\begin{array}{ccc}2& 0& -1\\ 5& 1& 0\\ 0& 1& 3\end{array}\right|=6-5=1$
$\therefore adj\left(adjA\right)=A$
$\therefore$ Assertion (A) is true.
Now $adj\left(adjA\right)=|A{|}^{(n-2)}A$
$\Rightarrow |adj\left(adjA\right)|=|A{|}^{n(n-2)}|A|=|A{|}^{(n-1{)}^2}$
$\therefore$ Reason (R) is also true but it is not the correct explanation of Assertion A.
Hence, option B.