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  • Which of the following statement is true?Option: 1 If A is hermitian matrix then, iA is a skew hermitian matrix, where i = √-1</d

Which of the following statement is true?

Option: 1

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


Option: 2

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


Option: 3

If A and B are hermitian matrices of same order, then AB - BA will be skew-hermitian ​​​​​​.


Option: 4

All of the above


Answers (1)

best_answer

 

 

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

A=\left[\begin{array}{cc}{a} & {b+i c} \\ {b-i c} & {d}\end{array}\right]

then iA=\left[\begin{array}{cc}{ai} & {bi- c} \\ {bi+ c} & {di}\end{array}\right]=\left[\begin{array}{cc}{ai} & {- c+bi} \\ { c+bi} & {di}\end{array}\right] which is skew hermitian matrix.

 

now, iA=\left[\begin{array}{cc}{ai} & {bi- c} \\ {bi+ c} & {di}\end{array}\right]=\left[\begin{array}{cc}{ai} & {- c+bi} \\ { c+bi} & {di}\end{array}\right]

 

i^2A=\left[\begin{array}{cc}{ai^2} & {bi^2- ci} \\ {bi^2+ ci} & {di^2}\end{array}\right]=\left[\begin{array}{cc}{-a} & {- b-ci} \\ { -b+ci} & {-d}\end{array}\right]=-Awhich is hermitian matrix.

hence option a and b is true.

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

A=\left[\begin{array}{cc}{a} & {i c} \\ {-i c} & {d}\end{array}\right]\;\;\;B=\left[\begin{array}{cc}{e} & {i g} \\ {-i g} & {h}\end{array}\right]\;\;\;

Now taking transpose 

A'=\left[\begin{array}{cc}{a} & {-i c} \\ {i c} & {d}\end{array}\right]\;\;\;B'=\left[\begin{array}{cc}{e} & {-i g} \\ {i g} & {h}\end{array}\right]\;\;\;

Now taking conjugate

A^{\theta}=\left[\begin{array}{cc}{a} & {i c} \\ {-i c} & {d}\end{array}\right]\;\;\;B^{\theta}=\left[\begin{array}{cc}{e} & {i g} \\ {-i g} & {h}\end{array}\right]\;\;\;

\\A\cdot B=\left[\begin{array}{cc}{a} & {i c} \\ {-i c} & {d}\end{array}\right]\cdot \left[\begin{array}{cc}{e} & {i g} \\ {-i g} & {h}\end{array}\right] \\\\A\cdot B=\left[\begin{array}{cc}{ae+cg} &\;\;\; {iag+ich} \\ {-iec-idg} & {gc+dh}\end{array}\right] \\\\A\cdot B=\left[\begin{array}{cc}{ae+cg} &\;\;\; {i(ag+ch)} \\ {-i(ec+dg)} & {gc+dh}\end{array}\right]

\\B\cdot A=\left[\begin{array}{cc}{e} & {i g} \\ {-i g} & {h}\end{array}\right] \cdot\left[\begin{array}{cc}{a} & {i c} \\ {-i c} & {d}\end{array}\right] \\\\B\cdot A=\left[\begin{array}{cc}{ae+gc} &\;\;\; {iec+igd} \\ {-iag-ihc} & {cg+dh}\end{array}\right]

\\A\cdot B-B\cdot A=\left[\begin{array}{cc}{0} &\;\;\; {i(ag+ch+ec+gd)} \\ {-i(ec+dg+ag+hc)} & {0}\end{array}\right]this skew hermitian matrix

 therefore, statement (c) is also correct

Hence correct option is (d)

 

Posted by

HARSH KANKARIA

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