If $P$ is a $3 \times 3$ matrix such that $P^T = 2P + I$, where $P^T$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ such that Options: (A) (a) $PX = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ (B) (b) $PX = X$ (C) (c) $PX = 2X$ (D) (d) $PX = -X$

### Analyzing the Setup The problem presents us with the matrix equation: $$P^T = 2P + I$$ To solve for $P$, we must utilize the fundamental property of the transpose operator, which is an involution. Specifically, for any matrix $P$, the identity $(P^T)^T = P$ holds true. ### The Master Equation We begin by taking the transpose of both sides of the original equation: $$(P^T)^T = (2P + I)^T$$ Applying the linearity property of the transpose, where $(A+B)^T = A^T + B^T$ and $(kA)^T = kA^T$, the equation simplifies to: $$P = 2P^T + I$$ Now, we possess a system of two linear equations involving $P$ and $P^T$: 1) $P^T = 2P + I$ 2) $P = 2P^T + I$ ### Solving the System We substitute the expression for $P^T$ from the first equation into the second equation: $$P = 2(2P + I) + I$$ Expanding the terms on the right side, we obtain: $$P = 4P + 2I + I$$ $$P = 4P + 3I$$ Rearranging the terms to isolate $P$ on one side: $$P - 4P = 3I$$ $$-3P = 3I$$ ### Final Result Dividing both sides by $-3$, we arrive at the final solution: **$P = -I$** This result indicates that $P$ is the negative identity matrix. Consequently, for any column vector $X$, the transformation yields $PX = -IX = -X$, which geometrically represents a reflection of the vector through the origin.

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Animated Solution for Mathematics - Matrices and Determinants: If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I$ , where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = x y z \neq = 000$ such that

Q: Wait, why can't we just assume $P^T = P$ because it is a matrix?

That is a common trap! Assuming $P^T = P$ is equivalent to assuming $P$ is a symmetric matrix. However, the problem does not state that $P$ is symmetric. We must derive the properties of $P$ strictly from the given equation $P^T = 2P + I$ without making any external assumptions.

Q: Is it mathematically valid to take the transpose of the entire equation?

Absolutely. If two matrices $A$ and $B$ are equal, then $A^T$ must equal $B^T$. This is a standard and powerful technique in linear algebra to generate a system of equations when you have a relationship involving a transpose.

Q: Why does $PX = -X$ lead to the correct option?

We found that $P = -I$. When we multiply this by any column vector $X$, we get $PX = (-I)X = -IX = -X$. This confirms that for any non-zero vector $X$, the transformation $P$ maps it to its negative.

Select Answer:

Visualized Solution

Analyze the Given Equation

Given equation: $P^{T} = 2 P + I$
$P$ is a $3 \times 3$ matrix and $I$ is the identity matrix.
We need to find the value of $P X$ for a non-zero column matrix $X$ .

Apply Transpose to Both Sides

Take the transpose on both sides: $(P^{T})^{T} = (2 P + I)^{T}$
Using the property $(A^{T})^{T} = A$ , the left side becomes $P$ .
Using $(A + B)^{T} = A^{T} + B^{T}$ and $I^{T} = I$ , the right side becomes $2 P^{T} + I$ .
New equation: $P = 2 P^{T} + I$

Substitute $P^{T}$ back into the Equation

We have two equations:
1. $P^{T} = 2 P + I$
2. $P = 2 P^{T} + I$
Substitute the value of $P^{T}$ from the first equation into the second:
$P = 2 (2 P + I) + I$

Simplify the Matrix Equation

Expand the brackets: $P = 4 P + 2 I + I$
Combine the identity matrix terms: $P = 4 P + 3 I$

Solve for Matrix $P$

Rearrange the terms to isolate $P$ : $4 P - P = - 3 I$
Simplify: $3 P = - 3 I$
Divide by $3$ : $P = - I$

Final Conclusion for $P X$

We need to evaluate $P X$ .
Substitute $P = - I$ : $P X = (- I) X$
Since multiplying any matrix by the identity matrix leaves it unchanged ( $I X = X$ ):
$P X = - X$
Therefore, the correct option is (d).

00:00 / 00:00

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The problem presents us with the matrix equation:

P^{T} = 2 P + I

To solve for

P

, we must utilize the fundamental property of the transpose operator, which is an involution. Specifically, for any matrix

P

, the identity

(P^{T})^{T} = P

holds true.

The Master Equation

We begin by taking the transpose of both sides of the original equation:

(P^{T})^{T} = (2 P + I)^{T}

Applying the linearity property of the transpose, where

(A + B)^{T} = A^{T} + B^{T}

and

(k A)^{T} = k A^{T}

, the equation simplifies to:

P = 2 P^{T} + I

Now, we possess a system of two linear equations involving

P

and

P^{T}

: 1)

P^{T} = 2 P + I

P = 2 P^{T} + I

Solving the System

We substitute the expression for

P^{T}

from the first equation into the second equation:

P = 2 (2 P + I) + I

Expanding the terms on the right side, we obtain:

P = 4 P + 2 I + I

P = 4 P + 3 I

Rearranging the terms to isolate

P

on one side:

P - 4 P = 3 I

- 3 P = 3 I

Final Result

Dividing both sides by

- 3

, we arrive at the final solution: $P = - I$

This result indicates that

P

is the negative identity matrix. Consequently, for any column vector

X

, the transformation yields

P X = - I X = - X

, which geometrically represents a reflection of the vector through the origin.

Common Traps & FAQs

Wait, why can't we just assume

P^{T} = P

because it is a matrix?

That is a common trap! Assuming

P^{T} = P

is equivalent to assuming

P

is a symmetric matrix. However, the problem does not state that

P

is symmetric. We must derive the properties of

P

strictly from the given equation

P^{T} = 2 P + I

without making any external assumptions.

Is it mathematically valid to take the transpose of the entire equation?

Absolutely. If two matrices

A

and

B

are equal, then

A^{T}

must equal

B^{T}

. This is a standard and powerful technique in linear algebra to generate a system of equations when you have a relationship involving a transpose.

Why does

P X = - X

lead to the correct option?

We found that

P = - I

. When we multiply this by any column vector

X

, we get

P X = (- I) X = - I X = - X

. This confirms that for any non-zero vector

X

, the transformation

P

maps it to its negative.

Animated Solution for Mathematics - Matrices and Determinants: If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I$ , where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = x y z \neq = 000$ such that

Select Answer:

Visualized Solution

Analyze the Given Equation

Apply Transpose to Both Sides

Substitute $P^{T}$ back into the Equation

Simplify the Matrix Equation

Solve for Matrix $P$

Final Conclusion for $P X$

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Master Equation

Solving the System

Final Result

Similar Questions

Let $P_{1} = I = 100010001, P_{2} = 100001010, P_{3} = 010100001, P_{4} = 001100010, P_{5} = 010001100, P_{6} = 001010100$ and $X = \sum_{k = 1}^{6} P_{k} 213102321 P_{k}^{T}$ where $P_{k}^{T}$ denotes the transpose of the matrix $P_{k}$ . Then which of the following options is/are correct?

Let $A = [a_{ij}]$ be $3 \times 3$ matrix such that $A 010 = 001$ , $A 413 = 010$ and $A 212 = 100$ , then $a_{23}$ equals:

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Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$ . If $P^{T}$ denotes the transpose of $P$ , then $M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T}$ is equal to

Let $A, B$ and $C$ be three $2 \times 2$ matrices with real entries such that $B = (I + A)^{- 1}$ and $A + C = I$ . If $B C = [1 - 1 - 5 2]$ and $C B [x_{1} x_{2}] = [12 - 6]$ , then $x_{1} + x_{2}$ is

Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in N$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P$ , then $α + β + γ - δ$ is equal to

Let $P = 139013001$ and $Q = [q_{ij}]$ be two $3 \times 3$ matrices such that $Q - P^{5} = I_{3}$ . Then $\frac{q _{21} + q _{31}}{q _{32}}$ is equal to:

For any $3 \times 3$ matrix $M$ , let $∣ M ∣$ denote the determinant of $M$ . Let $E = 12823133418$ , $P = 100001010$ and $F = 18231842133$ . If $Q$ is a nonsingular matrix of order $3 \times 3$ , then which of the following statements is (are) TRUE ?

The total number of matrices $A = 0 2 x 2 x 2 y y - y 1 - 1 1, (x, y \in R, x \neq = y)$ for which $A^{T} A = 3 I_{3}$ is :-

Select Answer:

Visualized Solution

Analyze the Given Equation

Apply Transpose to Both Sides

Substitute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PT back into the Equation

Simplify the Matrix Equation

Solve for Matrix .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P

Final Conclusion for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } PX

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Master Equation

Solving the System

Final Result

Similar Questions

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Substitute $P^{T}$ back into the Equation

Solve for Matrix $P$

Final Conclusion for $P X$

Let $A = [a_{ij}]$ be $3 \times 3$ matrix such that $A 010 = 001$ , $A 413 = 010$ and $A 212 = 100$ , then $a_{23}$ equals:

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Let $P = 1416014001$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$ , then $\frac{q _{31} + q _{32}}{q _{21}}$ equals

Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N = N M$ . If $P^{T}$ denotes the transpose of $P$ , then $M^{2} N^{2} (M^{T} N)^{- 1} (M N^{- 1})^{T}$ is equal to

Let $A, B$ and $C$ be three $2 \times 2$ matrices with real entries such that $B = (I + A)^{- 1}$ and $A + C = I$ . If $B C = [1 - 1 - 5 2]$ and $C B [x_{1} x_{2}] = [12 - 6]$ , then $x_{1} + x_{2}$ is

Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in N$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P$ , then $α + β + γ - δ$ is equal to

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The total number of matrices $A = 0 2 x 2 x 2 y y - y 1 - 1 1, (x, y \in R, x \neq = y)$ for which $A^{T} A = 3 I_{3}$ is :-