Question:

Let x be an n×1n \times 1 matrix. Let O and I be the zero, and identity matrices of order n, respectively. Define P=−xxTxTxP = - \frac{xx^T}{x^Tx} is the transpose of x. Then which of the following options is always CORRECT?

Updated On: Mar 10, 2025
  • P2−P=OP^2 - P = O
  • P2−P=IP^2 - P = I
  • P2+P=OP^2 + P = O
  • P2+P=IP^2 + P = I
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The Correct Option is C

Solution and Explanation

Since it is given that 
P=−xxTxTxP= - \frac{x x^{T}}{x^{T} x}
⇒xTxP=−xxT\Rightarrow x^{T} xP = -xx^{T}
On applying transpose both sides, we get 
PTxTx=−xTxP^{T}x^{T}x = -x^{T}x
⇒(PT+I)xTx=O\Rightarrow \left(P^{T} +I\right) x^{T}x = O
⇒PT+I=0\Rightarrow P^{T} + I =0
⇒PT=−I∴P=−I{∵IT=I}\Rightarrow P^T = -I \therefore P = -I \left\{ \because I^{T} = I\right\}
so P2+P=I−I=OP^{2} +P = I - I =O
⇒P2+P=O\Rightarrow P^{2} +P = O

Therefore, The Correct Answer is (C) P2 + P = O

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Concepts Used:

Transpose of a Matrix

The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”

The transpose matrix of A is represented by A’. It can be better understood by the given example:

A Matrix
A' Matrix
The transpose matrix of A is denoted by A’

Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.

Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.

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