Calculating the inverse of this matrix....?

Question

Ok I have no idea how to figure out the inverse of this 2x2 matrix:

9 4
5 2

im likee 15 so i really dont understand a lot of places on the internet that talk about inverses of matrices... actually all ive ever done is adding subtracting and multiplying them so can somebody explain to me how to do this ?
thanks :]

Answer 1

Three steps:

1) Find determinant
2) Reciprocate it
3) Find inverse matrix

Here are the details:

1) To find determinant, cross-multiply and subtract. So, in your case 9x2 - 5x4 = 18 - 20 = -2.

2) Reciprocated, -2 becomes -1/2

3) To find inverse matrix, there are two things to do. Change the signs of the 5 and 4, and swap the 9 and 2. So your new 2x2 matrix is written as

2 -4
-5 9

Place the -1/2 in front of it, and you are done.

By placing in front, this means you are multiplying, so you CAN distribute if you want to into all 4 numbers.

Answer 2

Since you seem to be inexperienced in dealing with matrices, let us proceed from a definition. The inverse of the matrix A is a matrix B such that AB = BA = I, where I is the "identity " matrix

1 0
0 1.

For your matrix you want to find a matrix

a b
c d

such that the product with your matrix is I. But equating the product with the corresponding entries in the matrix I gives us

(1) 9a + 4c = 1
(2) 9b + 4d = 0
(3) 5a + 2c = 0
(4) 5b + 2d = 1.

From (1) and (3) we find a = -1 and c = 5/2. From (2) and (4) we find b = 2 and d = -9/2. Therefore, your inverse is

-1 2
5/2 -9/2 .

What dwalon did is give you the general formula for the inverse of an invertible matrix. If the given matrix is

a b
c d

and det = ad - bc is not 0, the inverse is

d/det -b/det
-c/det a/det .