The minimum amount of framing Lorraine needs for the square painting is 31 inches.
We can use the Pythagorean theorem to calculate the length of the side of the square given the length of its diagonal. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
We know that the diagonal of the square painting is 44 inches, and since the diagonal of a square bisects its angles, the square can be divided into two congruent right-angled triangles. Therefore, the length of one side of the square can be calculated using the Pythagorean theorem:
44 inches^2 = x^2 + x^2
x^2 = 11*11
x = 11 inches
Since the square painting has a side length of 11 inches, and the framing material is sold by the inch, Lorraine will need a minimum of 11 inches of framing on each side, for a total of 44 inches.