# The pair of matrices that are inverses?

• Apr 16th 2012, 05:10 PM
jenbones
The pair of matrices that are inverses?
I need to find the pair of matrices that are inverses, but I don't fully understand how to inverse matrices.

P = [3 1_-4 0] q = [4 4_12 21]
R = [0 -1/4_1 3/4]
S = [7/12 -1/9_-1/3 1/9]
T = [4 -5 2_8 -1 3]
U = [-9 6 4_-5 -2 3]
V = [3 1_0 2_-4 5]

-------------------------------------------------

a. S and R
b. Q and R
c. P and Q
d. Q and S

if you could explain it to me that would be great, thanks.
• Apr 16th 2012, 05:25 PM
Daniiel
Re: The pair of matrices that are inverses?
You can either,

Multiply the four choices a,b,c,d out.

or find each of there determinants.

If B is the inverse of A then AB= I

So the answer will be what two matrix multiply together to give the identity matrix.

Finding their determinants you say B is the inverse of A and Det(A) = 2, then Det(A-1)=Det(B)=1/2

This means Det(A)Det(B)=1 which means A and B are inverses.

I'd probably multiply them all out its good practice
• Apr 16th 2012, 05:30 PM
jenbones
Re: The pair of matrices that are inverses?
Quote:

Originally Posted by Daniiel
You can either,

Multiply the four choices a,b,c,d out.

or find each of there determinants.

If B is the inverse of A then AB= I

So the answer will be what two matrix multiply together to give the identity matrix.

Finding their determinants you say B is the inverse of A and Det(A) = 2, then Det(A-1)=Det(B)=1/2

This means Det(A)Det(B)=1 which means A and B are inverses.

I'd probably multiply them all out its good practice

I'm also dealing with fractions, which is the main part of the confusion I am having with inverses
• Apr 16th 2012, 05:49 PM
Daniiel
Re: The pair of matrices that are inverses?
Oh okay, with fractions the calculation is a little bit more annoying, but exactly the same process.

For example a)

S = [7/12 -1/9_]

R = [0 -1/4_1 3/4]

SR =
7/12 -1/9 ) ( 0 -1/4
-1/3 1/9 ) ( 1 3/4

=
-1/9 (7/12)(-1/4) +(-1/9)(3/4) (can already tell its not I since top r ow isn't 1 0
1/9 (-1/3)(-1/4) + (-1/3)(3/4)

So S and R are not inverse,

alternatively

Det(S) = 1/36 and Det(R) = -1/4 showing that they are not inverses

If your still having trouble with fractions one thing you can do is take out factors,

so R = [0 -1/4_1 3/4] = 1/4 * [0 -1_4 3]

Then you simply multiply in the fraction after multiplying the matrix to another