Let x be an $n \times 1$ matrix. Let O and I be the zero, and identity matrices of order n, respectively. Define $P = - \frac{xx^T}{x^Tx}$ is the transpose of x.
Then which of the following options is always CORRECT?
- $P^2 - P = O$
- $P^2 - P = I$
- $P^2 + P = O$
- $P^2 + P = I$
The Correct Option is C
Solution and Explanation
Since it is given that
\(P= - \frac{x x^{T}}{x^{T} x}\)
\(\Rightarrow x^{T} xP = -xx^{T}\)
On applying transpose both sides, we get
\(P^{T}x^{T}x = -x^{T}x\)
\(\Rightarrow \left(P^{T} +I\right) x^{T}x = O\)
\(\Rightarrow P^{T} + I =0\)
\(\Rightarrow P^T = -I \therefore P = -I \left\{ \because I^{T} = I\right\}\)
so \(P^{2} +P = I - I =O\)
\(\Rightarrow P^{2} +P = O\)
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:
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.
Read More: Transpose of a Matrix