Only a square matrix - a matrix with an equal number of rows and columns - has an inverse. Therefore, a matrix with five rows and four columns cannot even have an inverse.

Let us examine each of the options:

This is the definition of an orthogonal matrix; it is always true.

This can be directly proved from the previous statment. If you subtitute the inverse for the transpose in the definition equation, it is still true.

The determinant of any orthogonal matrix is either 1 or -1. This statment can be proved in the following way:

The incorrect statment is . Consider an example matrix:

which has a transpose

M and its transpose are clearly not equal. However, if we multiply them, we can see that their product is the identity matrix and they are therefore orthogonal.

is an orthogonal matrix, if, by definition, , where is the conjugate transpose of . Also, for any square matrix , it holds that .

Let be orthogonal. Since

,

it follows that

Matrix multiplication is associative, so

By similar reasoning, it can be demonstrated that . is therefore unitary.

The trace of a square matrix is equal to the sum of its diagonal elements - the elements whose row number and column number are equal. Therefore, if and are matrices,

and

By the addition rule for finite sums,

Also, addition of matrices is elementwise, so

To find the norm, square each component, add, then take the square root:

The trace of a matrix is simply adding the entries along the main diagonal.

We need to get the matrix into reduced echelon form, and then count all the non all zero rows.

The rank is 2, since there are 2 non all zero rows.

Since is a matrix, . Since has three linearly independent columns, it must have a column space (and hence row space) of dimension , causing by the definition of rank. Hence.

The sum of the rank and the nullity of a matrix is equal to the number of columns in the matrix. Since the matrix in question has six columns, for the rank and the nullity to be equal, they must each be 3.