Binary Operation: Find Identity Element & Inverse

[sara_87]

Homework Statement

A binary operation is defined by:
the set of 2 x 2 matrices with real entries under matrix multiplication.
Find the identity element and the inverse.

Homework Equations

identity element: a$$\o$$e=e$$\o$$a=a

inverse: a$$\o$$b=b$$\o$$a=e

The Attempt at a Solution

I think that the identity element is a 2 x 2 matrix of zeroes with 1's along the diagonal. but how do i find the inverse b?
Thank you.

 

[Office_Shredder]

In general A will not have an inverse B such that AB = Identity. The brute force method is try multiplying a matrix with elements a,b,c,d with a matrix with elements e,f,g,h such that the resulting matrix is the identity. Find e,f,g,h in terms of a,b,c,d. The hint is that ad-bc is going to have to be non-zero for you to be able to solve this equation (because you have to divide by ad-bc)

 

[sara_87]

so if i do this, i get:
ae+bg=1
cf+dh=1
af+bh=0
ce+dg=0
why do i divide by ad-bc?

 

[Mark44]

'cause you're not done yet. You have to find the numbers e, f, g, and h in order to say what the inverse is for a given matrix A.

Here are a couple of 2 x 2 matrices, one of which has an inverse and the other doesn't.

A = [1 1; 0 2] (listed by rows)

B = [1 2; 3 6] (ditto)

 

[sara_87]

ok i see what you mean, i found e, f, g, and h:
e=d/(ad-bc)
f=-b/(ad-bc)
g=-c/(ad-bc)
h=a/(ad-bc)

Now, what if i had to find the identity element and inverse of a set of 2 x 2 matrices with non-zero determinant (under matrix multiplication).
Does this mean that no such 'b' exists?

 

[Office_Shredder]

If the determinant is non-zero (i.e. ad-bc =/= 0) then you pick your matrix (e,f,g,h) just as you found in your post. Hence in the set of all matrices with non-zero determinant, the identity matrix is the identity, and inverses exist just as you found. It's only when ad-bc=0 that you can't find an inverse