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MathMedAdv 2012 • Paper 2

Existence of Column Matrix for Given Transpose Matrix Relation

If PP is a 3×33 \times 3 matrix such that PT=2P+IP^T = 2P + I, where PTP^T is the transpose of PP and II is the 3×33 \times 3 identity matrix, then there exists a column matrix X=[xyz]≠[000]X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} such that

Options

A

PX=[000]PX = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}

B

PX=XPX = X

C

PX=2XPX = 2X

D

PX=−XPX = -X

Correct

Topics & Concepts

MatricesMatrices

Step-by-Step Solution

To find the relation satisfied by the column matrix X≠[000]X \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}, we begin with the given matrix equation: PT=2P+IP^T = 2P + I

Taking the transpose on both sides of the equation: (PT)T=(2P+I)T(P^T)^T = (2P + I)^T

Since (PT)T=P(P^T)^T = P and (2P+I)T=2PT+IT=2PT+I(2P + I)^T = 2P^T + I^T = 2P^T + I, we have: P=2PT+IP = 2P^T + I

Now, substitute the expression for PTP^T from the original equation into this result: P=2(2P+I)+IP = 2(2P + I) + I P=4P+2I+IP = 4P + 2I + I P=4P+3IP = 4P + 3I

Rearranging the terms gives: 3P+3I=O3P + 3I = O 3(P+I)=O3(P + I) = O P+I=O  ⟹  P=−IP + I = O \implies P = -I

Multiplying both sides by any non-zero column vector X=[xyz]≠[000]X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}: PX=(−I)X=−XPX = (-I)X = -X

Thus, there exists a non-zero column matrix XX such that PX=−XPX = -X.

Therefore, the correct option is (D).

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