A symmetric matrix M must follow the following condition:

We can find the transpose of A and compare to find x:

We can see that x must be equal to 7.

A matrix M is symmetric if it satisfies the condition:

We can find the transpose of P and see if it satisfies this condition:

Comparing the equations, we can see:

And so we can determine that matrix P is not symmetric.

A matrix M is symmetric if it satisfies the following condition:

The only matrix that satisfies this condition is:

Reversing the rows and columns of this matrix (finding the transpose) results in the same matrix.  Therefore, it is symmetric.

A skew-symmetric matrix is one whose transpose  is equal to its additive inverse .

 can be found by interchanging its rows with its columns:

Also, 

For  to be skew-symmetric, it must hold that 

That is, 

Using a property of logarithms:

 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 

.

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.

A Hermitian matrix  is equal to its conjugate transpose , which is the result of interchanging rows and columns, then changing entry to its complex conjugate.

For  to be true, all elements in corresponding positions must be equal. The diagonal elements are already equal, so examine the other elements. It must hold that 

and 

From both statements, it is necessary and sufficient to show 

and

For any , 

and 

Set  in both identities; the resulting statements are

and

,

precisely what is needed to be proved. It follows that  is indeed 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.

All three types of matrices are defined in terms of how  compares to its transpose.

 is symmetric if and only if  , so find , the transpose, by interchanging its rows and its columns:

, so  is symmetric.

 is skew-symmetric if and only if  . Find , the additive inverse of : 

, so  is not skew-symmetric.

 is Hermitian if and only if  , its conjugate transpose, so find  by replacing each entry in  with its complex conjugate:

, so  is not Hermitian.

A matrix is skew-symmetric if it is equal to the additive inverse of its transpose. Taking the transpose of each matrix by interchanging rows with columns:

 is skew-symmetric. 

 is not skew-symmetric.

 is not skew-symmetric.