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 –Hermitian matrix i.e. $a_{ij}=-\overline{a_{ji}}$ for all 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 $f\left(A\right)=\{\begin{array}{c}1Aishermitian\\ -1Aisskewhermitian\end{array}$
$f(A-A^{\theta })=$

• A. 1
• B. -1
• C. +1 any when f(A) >0
• D. -1 any when f(A) >0

Answer 1

The correct option is B -1

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 M is set of all hermitian or Skew–hermitian matrixes, be a function defined as
$f\left(a\right)=\{\begin{array}{cc}1& Aishermitian\\ -1& Aisskewhermitian\end{array}$
$X=A-A^{\theta }$ is a skew hermitian
${(\because X^{\theta }=(A-A^{\theta })}^{\theta }=A^{\theta }-A=-X\left)f\right(X)=1$