Question

If P is a 3 × 3 matrix such that $\sf P^T = 2P + I$ where I and $\sf P^T$ are matrices of order 3 × 3 and there exists $\bigg[\begin{array}{c} x & y & z \end{array} \bigg]$ such that :
1) PX = $\sf \{0\} _{3 \times 3}$
2) PX = 2X
3) PX = X
4) PX = - X ​

Answer 1

Answer:

Given that,

P' = 2P + I

where, I, P and P' are 3 × 3 matrices.

We are required to find the suitable solution matching with the options.

Property used in this question:

⇒ (A')' = A

That is, transpose of a matrix A' ( A transpose ) is equal to matrix A.

Now taking transpose on both sides of the given condition we get:

⇒ P' = 2P + I

⇒ (P')' = (2P + I)'

⇒ P = 2P' + I'

⇒ P = 2P' + I (Since I' = I) ...(Eqn. 1)

Now Substituting the value of P' = 2P + I in Eqn. (1) we get:

⇒ P = 2 ( 2P + I ) + I

⇒ P = 4P + 2I + I

⇒ P = 4P + 3I

Transposing I and P terms to their respective sides we get:

⇒ -3I = 4P - P

⇒ -3I = 3P

⇒ P = -I  (or) P = -1

Now multiplying X on both sides we get:

⇒ PX = -IX

⇒ PX = -(X)

Hence Option (4) is the correct relation.

Answer 2

Answer:

Question :-

$\leadsto$ If P is a 3 × 3 matrix such that $\sf P^T =\: 2P + I$ where I and $\sf P^T$ are matrices of order 3 × 3 are there exists $\sf \bigg[\begin{array}{c} x & y & z \end{array}\bigg]$ such that :

(1) $\sf PX =\: \{0\} _{3 \times 3}$

(2) $\sf PX =\: 2X$

(3) $\sf PX =\: X$

(4) $\sf PX =\: - X$

Given :-

• If P is a 3 × 3 matrix such that Pᵀ = 2P + I where I and Pᵀ are matrices of order 3 × 3 are there exists $\sf \bigg[\begin{array}{c} x & y & z \end{array} \bigg]$.

Solution :-

$\longmapsto$ $\sf {P}^{T} =\: 2P + 1$

$\implies$ $\sf {({P}^{T})}^{T} =\: {(2P + 1)}^{T}$

$\implies$ $\sf P =\: {2P}^{T} + I$

$\implies$ $\sf P =\: 2(2P + I) + I$

$\implies$ $\sf P =\: 4P + 2I + I$

$\implies$ $\sf P =\: 4P + 3I$

$\implies$ $\sf 3P =\: - 3I$

$\implies$ $\sf P =\: - I$

$\implies$ $\sf PX =\: - IX$

$\implies$ $\sf\bold{\red{P =\: - X}}$

Hence, the correct options is option no 4) PX - X.

Extra Information :-

➲ Matrix :

• Matrix was discovered by James Sylvester.
• Matrix is a way of arrangements of number, expressions and symbols in different rows and columns.