Question

How do you find the inverse of `A=``((-3, -1), (-7, 8))`?

Answer 1

`A^(-1)=` `((-8/31, -1/31), (-7/31, 3/31))`

Explanation:

`A=` `((-3, -1), (-7, 8))`

`Cof(A)=` `((8, 7), (1, -3))`

`Adj(A)=` `((8, 1), (7, -3))`

`Det(A)=(-3)*8-(-1)(-7)=-31`

Thus, `A^(-1)=(Adj(A))/(Det(A))`

`A^(-1)=` `((-8/31, -1/31), (-7/31, 3/31))`

Answer 2

Answer: `[(-8/31,-1/31),(-7/31,3/31)]`

Explanation:

An easy way to find the inverse of a 2x2 square matrix is to apply the following formula*:
Let `A=[(a,b),(c,d)]`, then
`A^(-1)=1/(det(A))[(d,-b),(-c,a)]`

For this problem, we have
`A=[(-3,-1),(-7,8)]`, so `a=-3,b=-1,c=-7,d=8`

We can find the determinant:
`det(A)=ad-bc=-24-7=-31`

Plugging in our values, we have:
`A^(-1)=1/(-31)[(8,1),(7,-3)]=[(-8/31,-1/31),(-7/31,3/31)]`, which is our answer

*Note: This formula only works for 2x2 matrices and does not work for larger matrices