Thread: 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]

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answer choices:

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.

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

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

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