Question

A square matrix A such that $A^{\theta }=A$ is called Hermitian matrix i.e. $a_{ij}=\overline{a_{ji}}$ for all values of i and j and a square matrix A such that $A^{\theta }=-A$ is called skew -Hermitain matrix i.e. $a_{ij}=-\overline{a_{ji}}$ for all values of i and j. where is conjugate transpose matrix.
Let $f:M\to \{1,-1\},M$ is set of all hermitian or Skew–hermitian matrixes, be a function defined as
$f\left(A\right)=\{\begin{array}{cc}1& Aishermitian\\ -1& Aisskewhermitian\end{array}$
Let $Y=A^n$

• A. if f(Y)=1 then f(A)=1
• B. if f(Y)= –1 then f(A) = –1
• C. f(Y).f(A)=1 if n is odd
• D. f(Y).f(A)=1 if n is even

Answer 1

The correct option is C f(Y).f(A)=1 if n is odd

A square matrix A such that $A^{\theta }=A$ is called Hermitian matrix i.e. $a_{ij}=\overline{ji}$ for all values of i and j and a square matrix A such that $A^{\theta }=-A$ is called skew -Hermitian matrix i.e. $a_{ij}=-\overline{a_{ji}}$ for values of i and j. Where $A^{\theta }$ is conjugate transpose matrix.
Let $f:M\to \{1,-1\}$,M is set of all hermitian or Skew–hermitian matrices, be a function defined as
$f\left(A\right)=\{\begin{array}{c}1Aishermitian\\ -1Aisskewhermitian\end{array}$
$Y^{\theta }=(A^n{)}^{\theta }=(A^{\theta }{)}^n=\{\begin{array}{cc}{A}^n=Y& ifAishermitian\\ (-A{)}^n=\{\begin{array}{c}YifAisskewhermitianandniseven\\ -YIfAisskewhermitianandnisodd\end{array}& \end{array}$