Question

Assume that AT = A-1 and BT = B-1. Show that (AB)T =

(AB)-1.Assume that AT = A-1 and BT = B-1. Show that (AB)T =
(AB)-1.

Answer 1

To show that (AB)T = (AB)-1, we need to prove that the transpose of the product AB is equal to the inverse of the product AB. By expanding and using the properties of matrix transpose and inverse, we can show that (AB)T = (AB)-1.

Step-by-step explanation:

To show that (AB)T = (AB)-1, we need to prove that the transpose of the product AB is equal to the inverse of the product AB.

Let's start by expanding the transpose of AB:

(AB)T = BTAT

Next, let's expand the inverse of AB:

(AB)-1 = B-1A-1

Since it's known that AT = A-1 and BT = B-1, we can rewrite the above equation as:

(AB)T = BTAT = B-1A-1 = (AB)-1

Therefore, we have shown that (AB)T = (AB)-1.