A matrix is singular - that is, not having an inverse - if and only if one of its eigenvalues is 0. Since 0 is not an element of its eigenvalue set,  is nonsingular.

A matrix is singular - that is, not having an inverse - if and only if one of its eigenvalues is 0. This is seen to be the case.

 is singular, so the matrix must have 0 as an eigenvalue. 

Let  be the other two eigenvalues. The sum of the eigenvalues of a matrix is equal to its trace, so 

and 

or

It follows that one eigenvalue must be 0, and the other two must be additive inverses.

, being a singular matrix, must have 0 as an eigenvalue; it also has  as an eigenvalue. Being , it will have one more; call this eigenvalue . 

The sum of the eigenvalues of a matrix is equal to its trace, so 

The set of eigenvalues is . The eigenvalues of a matrix are the solutions of its characteristic (polynomial) equation, which, as a consequence, is

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of by adding the alternating products of each entry in any row or column and the corresponding adjoint. The third column is the easiest to do this with:

Since , 0 is not an eigenvalue of .

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of by adding the alternating products of each entry in any row or column and the corresponding adjoint. The first row is the easiest to do this with:

Since ,  has zero as an eigenvalue.

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of in terms of  by taking the product of the main diagonal elements and subtracting the product of the other two:

Set this equal to 0 and solve for :

.

A necessary and sufficient condition for a number to be an eigenvalue of is for

to be true. Therefore, first, find ; this is

Set the determinant of this matrix, which is found by taking the product of the main diagonal elements and subtracting the product of the other two, equal to 0:

The eigenvalues of a matrix are the zeroes of its characteristic equation.

We can try to extract one or more zeroes using the Rational Zeroes Theorem, which states that any rational zero must be the positive or negative quotient of a factor of the constant and a factor of the leading coefficient. Since the constant is 57, and the leading coefficient of the polynomial is 1, any rational zeroes must be one of the set divided by 1, with positive or native numbers taken into account. Thus, any rational zeroes must be in the set

By trial and error, it can be determined that 3 is a solution of the equation:

True.

It follows that is a factor. Divide by ; the quotient can be found to be , which is prime.

The characteristic equation is factorable as

We already know that  is an eigenvalue; the other two eigenvalues are the zeroes of , which can be found by way of the quadratic formula:

An eigenvalue is a dominant eigenvalue if its absolute value is strictly greater than those of all other eigenvalues. Calculate the absolute values of all three eigenvalues:

Therefore,

.

Since no single eigenvalue has absolute value strictly greater than that of the other two, the matrix has no dominant eigenvalue.

The eigenvalues of a matrix are the zeroes of its characteristic equation.

can be solved as follows:

First, factor the polynomial by using the substitution . The equation can be changed to

The polynomial can be factored, using the "reverse FOIL" method, to

Substituting back:

Factoring further:

The zeroes of the polynomial are the zeroes of the individual factors, which can be calculated to be

An eigenvalue is a dominant eigenvalue if its absolute value is strictly greater than those of all other eigenvalues. The absolute values of all three eigenvalues are:

Since no single eigenvalue has absolute value strictly greater than that of the other three, the matrix has no dominant eigenvalue.