Question

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

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

Answer 1

Answer: (D)

$|A d j P|=1$

Answer 2

PT=aP+(a−1)I
⇒P=aPT+(a−1)I
⇒PT−P=a(P−PT)
⇒P=PT, as a=−1
Now, P=aP+(a−1)I
⇒P=−I⇒∣P∣=1
⇒∣AdjP∣=1
So, the correct option is (D) : $|A d j P|=1$