If P is a 3 × 3 real matrix such that P T = a P + ( a - 1 ) I , where a 1 , then [2023]

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Solution

Q.
If P is a $3 \times 3$ real matrix such that $P^{T} = a P + (a - 1) I,$ where $a > 1$ , then [2023]

1 $| Adj P |$ $> 1$

2 $| Adj P |$ $= \frac{1}{2}$

3 P is a singular matrix

4 $| Adj P |$ $= 1$

Ans.
(4)

$P is a 3 \times 3 real matrix$

$∴ P^{T} = a P + (a - 1) I, a > 1 \Rightarrow {(P^{T})}^{T} = a P^{T} + (a - 1) I$

$\Rightarrow P = a P^{T} + (a - 1) I \Rightarrow P = a (a P + (a - 1) I) + (a - 1) I$

$\Rightarrow P = a^{2} P + a (a - 1) I + (a - 1) I$

$\Rightarrow P - a^{2} P = (a - 1) I (a + 1) = (a^{2} - 1) I$

$\Rightarrow P (1 - a^{2}) = (a^{2} - 1) I$

$\Rightarrow P (1 - a^{2}) - (a^{2} - 1) I = 0 \Rightarrow (1 - a^{2}) (P + I) = 0$

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

$∴$ $| Adj P |$ $= {(- 1)}^{2} = 1$

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