A square matrix  is defined to be skew-symmetric if its transpose  - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

.

Interchanging rows with columns in , we see that if 

then 

Also, by changing each entry in  to its additive inverse, we see that 

Setting the two equal to each other, we see that:

The non-diagonal elements - all constants - are all equal. Looking at the diagonal elements, we see that it is necessary and sufficient for ; that is,  must be its own additive inverse. The only such number is 0, so .

Let  be a three-by-three matrix - this reasoning extends to matrices of any size.

Let

 is the transpose of the matrix, which is formed when its rows are interchanged with its columns; this is

Subtract elementwise:

A matrix is symmetric if and only it is equal to its transpose; it is skew-symmetric if and only if it is equal to the additive inverse of its transpose. Interchanging rows and columns in , we see that

.

Each element in  is the additive inverse of the corresponding element in , so 

,

making  a skew-symmetric matrix.

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.

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.

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.