### 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, $P=aP+(a-1) I$

$\Rightarrow P=-I \Rightarrow|P|=1$

$\Rightarrow|adj P|=1$