Question

Find the product of AB and BA. (See picture) Thanks?!!

Answer 1

See below:

Explanation:

You can multiply a matrix `A` by a matrix `B` if a row of `A` is as long as a column of `B`

To find the 1,1 element, for example, of the product matrix `AB`, we have to take the first row of `A`, and the first column of `B`, multiply them term by term and finally add up.

This goes for all the terms - to find the `i,j`-th element of `AB`, we need to do this with the `i`-th row of `A` and the `j`-th column of `B`.

In this example the matrices are

`A = [(4,-3),(-3,1)],qquadqquad B = [(-1/5, -3/5),(-3/5,-4/5)]`

So, to find the 1,1 th element of `AB` we need to take

• the first row of `A` : `((4,-3))`, and
• the first column of `B` : `((-1/5),(-3/5))`

and form the sum

`4 times (-1/5)+(-3) times (-3/5)=(-4+9)/5=1`

So, the 1,1 element of `AB` in this case is 1.

Similarly, we can find the 1,2 element to be

`4 times (-3/5)+(-3) times (-4/5)=(-12+12)/5=0`

Proceeding in this way, we can find the other elements, leading to

`AB = [[1,0],[0,1]]`

A similar calculation works for `BA` (this time you have to take rows from `B` and columns from `A`), and the result is

`BA = [[1,0],[0,1]]`

So, both `AB` and `BA` are equal to the `2 times 2` identity matrix and so, `B` is inverse to `A`.