#### Which of the following statement is true?Option: 1 If A is hermitian matrix then, iA is a skew hermitian matrix, where i = √-1Option: 2 If A is a skew-hermitian matrix then iA is a hermitian matrix, where i = √-1Option: 3 If A and B are hermitian matrices of same order, then AB - BA will be skew-hermitian ​​​​​​.Option: 4 All of the above

Properties of hermitian and skew-hermitian matrices -

Properties of hermitian and skew-hermitian matrices

ii) If A is hermitian matrix then:

iA is a skew hermitian matrix, where i = √-1

Proof: we need to show (iA)? = -iA

(iA)?= A?i? = A? (-i) = -iA?

Since A is hermitian so A? = A

Hence we have

-iA? = -iA. Proved.

iii) if A is a skew-hermitian matrix, then:

iA is a hermitian matrix, where i = √-1

Proof: we need to show (iA)? = iA

(iA)? = A?i? = A?(-i)

A?(-i) = Ai = iA   (since A is skew-hermitian, so A? = -A)

iv) if A and B are hermitian matrices of the same order, then

d.  AB - BA will be skew-hermitian

Proof: we need to show (AB-BA)* = -(AB-BA)

(AB-BA)* = (AB)* -  (BA)* = B*A* - A*B* = BA - AB = -(AB - BA)

Using A? = A and B? = B, proved.

-

Let A be a matrix of order 2x2

then  which is skew hermitian matrix.

now,

which is hermitian matrix.

hence option a and b is true.

Let A and B be a hermitian matrix of order 2x2

Now taking transpose

Now taking conjugate

this skew hermitian matrix

therefore, statement (c) is also correct

Hence correct option is (d)

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