+50 If A and B are matrices and x and y are vectors such that they aren't multiple of each other and:
Ax = y, Ay = x + 2y and
Bx = x + y, By = 2y.
There are scalars a, b, c, and d that satisfy:
(AB)x = ax + by and
(BA)x = cx + dy.
How do I find a, b, c, d?
+174 We can use the given information about A and B acting on x and y to find the expressions for (AB)x and (BA)x.
Finding (AB)x:
We know (AB)x = B(Ax) since matrix multiplication is associative.
From the givens, Ax = y. Substitute: (AB)x = B(y).
We are also given By = 2y. Substitute: (AB)x = B(2y) = 2(By) = 2(2y) = 4y. Therefore, we can write (AB)x = 4y. This translates to a = 0 and b = 4.
Finding (BA)x:
Similar to (AB)x, we know (BA)x = A(Bx).
From the givens, Bx = x + y. Substitute: (BA)x = A(x + y).
We are also given Ay = x + 2y. Substitute: (BA)x = A(x + y) = Ay - 2y = (x + 2y) - 2y = x. Therefore, we can write (BA)x = x. This translates to c = 1
and d = 0.
In conclusion, the scalars that satisfy the equations are:
a = 0
b = 4
c = 1
d = 0
This means:
(AB)x = 4y
(BA)x = x
