A matrix can be identified as symmetric, skew-symmetric, or Hermitian, if any of these, by comparing the matrix to its transpose, the result of interchanging rows with columns.

The transpose of is

A matrix is symmetric if and only if . This can be seen to not be the case.

A matrix is skew-symmetric if . Taking the additive inverse of each entry in , it can be seen that

.

is therefore skew-symmetric.

A matrix is Hermitian if it is equal to its conjugate transpose . Find this by changing each entry in to its complex conjugate:

is also Hermitian.

The answer to this question can be found by first comparing

and

.

Note that .

Also note that for all ,

and

Setting , we get that

and

.

It follows that

,

and that can be rewritten as

A matrix can be identified as symmetric, skew-symmetric, or Hermitian, if any of these, by comparing the matrix to its transpose, the result of interchanging rows with columns. The transpose of is

Each entry in is equal to the additive inverse of the corresponding entry in ; that is, .This identifies as skew-symmetric by definition.

is the transpose of - the result of interchanging the rows of with its columns. is the conjugate transpose of - the result of changing each entry of to its complex conjugate. Therefore, if

,

then

.

If is skew-symmetric, then . But if were symmetric, then . Both conditions would only hold if was the zero matrix, which is not always the case.

is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

Therefore, first, take the transpose of :

Obtain the conjugate transpose by changing each element to its complex conjugate:

Now find the additive inverse of this by changing each entry to its additive inverse:

For , or,

i

It is necessary and sufficient that the two equations

and

These conditions are equivalent, so

makes skew-Hermitian.

A symmetric matrix is one that equals its transpose. This means that a symmetric matrix can only be a square matrix: transposing a matrix switches its dimensions, so the dimensions must be equal. Therefore, the option with a non square matrix, 2x3, is the only impossible symmetric matrix.