Solveeit Logo
Login

Question

Question: A skew-symmetric matrix \(M\) satisfies the relation \({M^2} + I = 0\), where \(I\) is the unit of m...

A skew-symmetric matrix MM satisfies the relation M2+I=0{M^2} + I = 0, where II is the unit of matrix. The MM′MM' is equal to
A. IA.{\text{ }}I
B. 2IB.{\text{ }}2I
C. −IC.{\text{ }} - I
D.D. None of these

Explanation

Solution

Hint: - Use the property of orthogonality along with the property of skew symmetric matrix.

Given M2+I=0{M^2} + I = 0
⇒M2=−I\Rightarrow {M^2} = - I
We know that,
Property of orthogonal matrix M2=I{M^2} = I, if MM is of odd order and
M2=−I{M^2} = - I if MM is of even order.
Hence, MM is an orthogonal matrix of even order.
By the definition of orthogonal matrix, the product of a square matrix and its transpose gives an identity matrix. So MM′=IMM' = I
Hence, the correct option is AA .

Note: - In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. One way to express this is where the transpose of Q is and is the identity matrix. The property of orthogonal matrices must be remembered in order to solve such theoretical questions.