This equation represents an ellipse. To determine the center, major and minor axes, and eccentricity of the ellipse, we can rewrite the equation in standard form.

First, let's rearrange the equation:
x^2 - 6xy + 13y^2 = 100

To complete the square, we need to find the values that will make the left side of the equation a perfect square trinomial. We can do this by taking half of the coefficient of xy and squaring it:

(-6/2)^2 = 9

Now, add and subtract 9x^2 to the left side of the equation:
x^2 - 6xy + 9x^2 + 13y^2 = 100 + 9x^2

Combine like terms:
10x^2 - 6xy + 13y^2 = 100 + 9x^2

Now, factor the left side of the equation:
(10x^2 - 6xy + 13y^2) = (10x^2 - 6xy + 9x^2 + 4x^2 + 13y^2)

Simplify:
(10x^2 - 6xy + 13y^2) = (19x^2 - 6xy + 13y^2)

Now, rewrite the equation:
19x^2 - 6xy + 13y^2 = 100 + 4x^2

Combine like terms:
15x^2 - 6xy + 13y^2 = 100

Divide both sides by 100 to simplify:
(15x^2 - 6xy + 13y^2)/100 = 1

Now, we can rewrite the equation in standard form:
(15x^2)/100 - (6xy)/100 + (13y^2)/100 = 1

Simplify:
(3x^2)/20 - (3xy)/50 + (13y^2)/100 = 1

Now, we can determine the center, major and minor axes, and eccentricity of the ellipse.

The center of the ellipse is (0, 0) since the x and y terms are squared and have the same coefficient.

The major axis is the longer axis of the ellipse, and the minor axis is the shorter axis. To determine their lengths, we can find the square roots of the denominators of the x^2 and y^2 terms:

Major axis: sqrt(20) = 2sqrt(5)
Minor axis: sqrt(100) = 10

The eccentricity of the ellipse can be determined using the formula:
eccentricity = sqrt(1 - (b^2/a^2))

where a is the length of the major axis and b is the length of the minor axis.

eccentricity = sqrt(1 - (10^2)/(2sqrt(5))^2)
eccentricity = sqrt(1 - 100/20)
eccentricity = sqrt(1 - 5)
eccentricity = sqrt(-4)

Since the square root of a negative number is not defined in the real number system, the eccentricity is undefined.

Therefore, the center of the ellipse is (0, 0), the major axis has a length of 2sqrt(5), the minor axis has a length of 10, and the eccentricity is undefined.