To find the value of k, we need to determine the scaling factor applied by transformation M. When a 2D shape with area k is transformed using a matrix transformation, the area of the image will be affected by the determinant of the transformation matrix.
Let M be the transformation matrix, and let A be the original 2D shape with area k. The image of A, denoted as A', is obtained by applying the transformation M to A. The area of A' is given by:
Area of A' = |det(M)| * Area of A
In this case, you mentioned that the image has an area of 24, so we have:
Area of A' = 24
Area of A = k
Now, we can rearrange the formula to solve for k:
k = (1/|det(M)|) * Area of A'
To find the value of k, we need to know the determinant of the transformation matrix M. If you have the transformation matrix M, you can calculate the determinant, and then use it to find the value of k using the formula above.