# A skew-symmetric matrix $M$ satisfies the relation ${M^2} + I = 0$, where $I$ is the unit of matrix. The $MM'$ is equal to

$A.{\text{ }}I$

$B.{\text{ }}2I$

$C.{\text{ }} - I$

$D.$ None of these

Last updated date: 23rd Mar 2023

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Answer

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

Given ${M^2} + I = 0$

$ \Rightarrow {M^2} = - I$

We know that,

Property of orthogonal matrix ${M^2} = I$, if $M$ is of odd order and

${M^2} = - I$ if $M$ is of even order.

Hence, $M$ 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 $MM' = I$

Hence, the correct option is $A$ .

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.

Given ${M^2} + I = 0$

$ \Rightarrow {M^2} = - I$

We know that,

Property of orthogonal matrix ${M^2} = I$, if $M$ is of odd order and

${M^2} = - I$ if $M$ is of even order.

Hence, $M$ 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 $MM' = I$

Hence, the correct option is $A$ .

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.

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