What am I doing wrong in this matrix

I was told to find the inverse of the matrix A= ( -2 3 , -4 5) (The comma separates the two rows)

So I found the determinant to be -22 and altered the matrix to : ( 5 -3 , 4 -2)
and divided each by -22.
When double checking my answer by multiplying the inverse matrix by the original matrix, instead of getting the identity matrix I got (-1/11 0 , 0 -1/11)

Is that right, or am I doing something wrong?

The determinant is not -22, it's 2.

(-2)(5) - (3)(-4) = -10 + 12 = 2.

This messed everything up.

Remember it's diagonal product MINUS off diagonal product.

You made a mistake in computing the determinant
It is (-2)(5)-(3)(-4)
=(-10)+(12)
=2

With this change, you can verify that
5/2 -3/2
2 -1

is the inverse of A

5 . -3
4 . -2

determinant = - 10 - (-12) = + 2

The determinant is not -22. Note that detA = (ad-bc) = (-2)*5 - 3*(-4) = -10 + 12 = 2

Thus the inverse is 1/2 (5 -3, 4 -2) or (5/2 -3/2, 2 -1)