SATHEE: Matrices and Determinants 3 Question 7

Matrices and Determinants 3 Question 7

9. 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 $x 0$ there exists a column matrix, $X =$

$[\begin{matrix} x \\ y \\ z \end{matrix}]$ $\neq$ $[\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that

(a) $P X =$ $[\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

(b) $P X = X$

(d) $P X = - X$

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Answer:

Correct Answer: 9. (d)

Solution:

Given, $P^{T} = 2 P + I$

$\begin{array}{ll} ∴ & {(P^{T})}^{T} = (2 P + I)^{T} = 2 P^{T} + I \\ \Rightarrow & P = 2 P^{T} + I \\ \Rightarrow & P = 2 (2 P + I) + I \\ \Rightarrow & P = 4 P + 3 I or 3 P = - 3 I \\ \Rightarrow & P X = - I X = - X \end{array}$

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Matrices and Determinants 3 Question 7

9. If P is a 3×3 matrix such that PT=2P+I, where PT is the transpose of P and I is the 3×3 identity matrix, then x0 there exists a column matrix, X=

Answer:

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9. 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 $x 0$ there exists a column matrix, $X =$