Final answer:
Matrices A and B are inverse matrices of each other as their products AB and BA both result in the identity matrix.
Step-by-step explanation:
The question requires us to perform matrix multiplication of two given matrices A and B, and then analyze their relationship. For two matrices A and B, where A=[[3,5],[2,4]] and B=[[2,-2.5],[-1,1.5]], the products AB and BA are calculated as follows:
To compute AB, we multiply each element of the rows of A by the corresponding element of the columns of B, and then sum those products for each position in the resulting matrix:
AB = [[(3*2 + 5*(-1)), (3*(-2.5) + 5*1.5)], [(2*2 + 4*(-1)), (2*(-2.5) + 4*1.5)]]
= [[6 - 5, -7.5 + 7.5], [4 - 4, -5 + 6]]
= [[1, 0], [0, 1]]
To compute BA, we apply the same process, multiplying each element of the rows of B by the corresponding elements of the columns of A:
BA = [[(2*3 + -2.5*2), (2*5 + -2.5*4)], [(-1*3 + 1.5*2), (-1*5 + 1.5*4)]]
= [[6 - 5, 10 - 10], [-3 + 3, -5 + 6]]
= [[1, 0], [0, 1]]
After computing AB and BA, we observe that both products resulted in the identity matrix, which indicates a certain relationship known as inverse matrices. Matrices A and B are inverses of each other because when they are multiplied in either order, the result is the identity matrix.