The transposition of a matrix switches the rows and the columns, so the number of rows in the original matrix is equal to the number of columns in the transpose, and vice versa. Therefore, a matrix with six rows and seven columns has seven rows and six columns.

The transpose of a matrix switches the rows and the columns. Therefore, the first column of  has the same entries, in order, as the first row of , and so forth. Since 

it follows that

The transpose of the sum of two matrices is indeed equal to the sum of their transposes. Let us look at the two-by-two case - this reasoning can be generalized.

 and 

The transpose of a matrix switches the rows and the columns. Therefore, the first column of  has the same entries, in order, as the first row of , and so forth. Therefore, 

 and . 

Matrices are added term by term, so

, the transpose of the sum of the original matrices, can be found by first adding the matrices termwise:

Take the transpose of the sum:

Indeed, .

Every matrix has a transpose regardless of the value of its determinant; finding the transpose is a matter of repositioning the elements - a concept independent of determinant - so that its rows are the columns of the transpose, and vice versa.

Since  is a two-by-two matrix, we will let

,

where , , , and  represent real values.

The transpose of a matrix switches the rows and the columns. Therefore, the first column of  has the same entries, in order, as the first row of , and so forth. Therefore,

and

Add two matrices termwise:

Since 

or, equivalently,

The following must hold:

, 

or

,

or 

and 

,

or, equivalently,

.

Therefore, any matrix of the form

for some real  would make the statement correct. The only choice matching this pattern is the matrix

.

A square matrix is nonsingular - that is, it has an inverse - if and only if its determinant is 0.

Let   a column matrix. Then the transpose is equal to the row matrix 

.

The product of a row matrix and a column matrix, each with the same number of elements, is a one-by-one matrix whose only element is the sum of the squares of the elements, so

The determinant of a one-by-one matrix is its only entry, so  is a square matrix with a nonzero determinant. This proves that  can be nonsingular for some non-square .

Now, let , the two-by-two (or any other) identity matrix.  is a nonsingular square matrix, and , so

.

This proves that  can be nonsingular for some square .

Consequently, if , it cannot be determined whether or not  is a nonsingular square matrix.

The identity matrix of any dimension serves as a counterexample that proves the statement false. Examine the three-by-three identity

 is an upper triangular matrix - all of its elements above its main diagonal are zeroes. The transpose of  - the matrix formed by interchanging rows with columns - is  itself. Therefore,  is an upper triangular matrix whose transpose is also upper triangular.

Let  be a three-by-three matrix; this reasoning extends to square matrices of all sizes.

 is a lower triangular matrix, so all of the entries above its main (upper left to lower right) diagonal are zeroes; that is, 

.

, the transpose of , is the matrix formed by switching rows with columns, so

.

However,  is lower triangular also; as a consequence, 

,

and

.

This demonstrates that  must be a diagonal matrix - one with only zeroes off its main diagonal.