Matrices Question 11
Question 11 - 30 January - Shift 2
If $P$ is a $3 \times 3$ real matrix such that $P^{T}=aP+(a-1) I$, where $a>1$, then
(1) $P$ is a singular matrix
(2) $\mid$ adj $P \mid>1$
(3) $\mid$ adj $P \lvert,=\frac{1}{2}.$
(4) $|adj P|=1$
Show Answer
Answer: (4)
Solution:
Formula: Properties of Adjoint of a Matrix, Properties of Scalar Multiplication, Properties of Transpose of matrix
$P^{T}=aP+(a-1) I$
$\Rightarrow P=aP^{T}+(a-1) I$
$\Rightarrow P^{T}-P=a(P-P^{T})$
$\Rightarrow P=P^{T}$, as $a \neq-1$
Now, $P=aP+(a-1) I$
$\Rightarrow P=-I \Rightarrow|P|=1$
$\Rightarrow|adj P|=1$
