Question

If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then

• A. $|Adj P| >1$
• B. $|A d j P|=\frac{1}{2}$
• C. $P$ is a singular matrix
• D. $|A d j P|=1$

Answer 1

Answer: (D)

$ P ^{ T }= aP +( a -1) I $
$ \Rightarrow P =a P ^{ T }+( a -1) I $
$ \Rightarrow P ^{ T }- P = a \left( P - P ^{ T }\right) $
$ \Rightarrow P = P ^{ T }, \text { as } a \neq-1 $
$ \text { Now, } P = aP +( a -1) I $
$ \Rightarrow P =- I \Rightarrow| P |=1 $
$ \Rightarrow|\text{Adj} P |=1$