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Matrices system

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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?

Apr 15, 2024

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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

Apr 15, 2024