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Question
AnswersA 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
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.