Answer:
(1/2) {ad - cb} both expressions
Explanation:
Notice that the area of the triangle in question can be calculated as the area of the larger right angle triangle OSQ MINUS the area of the smaller right angle triangle OPR, and MINUS the area of the trapezoid RSQP.
Given the coordinates of the given points, the formulas for area of triangles and of trapezoid, we have the following area subtractions:
1/2 (a + c) (b +d) - 1/2 a* b - 1/2 c (b + b + d)=
= (1/2) { ab + ad + cb + cd - ab - 2 cb -cd } =
= (1/2) {ad - cb}
On the other hand the matrix expression is giving you 1/2 of the determinant of the given matrix such determinant is: ad - cb
Then we see that the equality is correct since (1/2) of the determinant is also :
(1/2) {ad - cb}