Question

How do you find the inverse of `A=``((1, 2), (5, 3))`?

Answer 1

`A^(-1) = ((-3/7, 2/7),(5/7,-1/7))`

Explanation:

Given any invertible matrix `A`, we can find `A^(-1)` by creating the augmented matrix `(A|I)` and then performing elementary row operations to change the initial matrix into the identity matrix. The result will be `(I|A^(-1))`

`(A|I) = ((1,2,|,1,0),(5,3,|,0,1))`

`R_2 - 5R_1->((1,2,|,1,0),(0,-7,|,-5,1))`

`-1/7R_2->((1,2,|,1,0),(0,1,|,5/7,-1/7))`

`R_1-2R_2->((1,0,|,-3/7,2/7),(0,1,|,5/7,-1/7)) = (I|A^(-1))`

Thus we have `A^(-1) = ((-3/7, 2/7),(5/7,-1/7))`

Verifying this, we find that `A A^(-1) = I`