Question

How do you find the inverse of `((5, 2), (-1, a))`?

Answer 1

Use the formula for the inverse of a `2xx2` matrix to find:

`((5,2),(-1,a))^(-1) = ((a/(5a+2),-2/(5a+2)),(1/(5a+2),5/(5a+2)))`

Explanation:

In general the inverse of a `2xx2` matrix is given by the formula:

`((a,b),(c,d))^(-1) = 1/abs((a,b),(c,d)) ((d,-b),(-c,a))`

In our example, let's calculate the determinant first:

`abs((5,2),(-1,a)) = 5a+2`

So provided `a != -2/5` the determinant is non-zero and the inverse matrix can be written:

`((5,2),(-1,a))^(-1) = 1/(5a+2) ((a,-2),(1,5))=((a/(5a+2),-2/(5a+2)),(1/(5a+2),5/(5a+2)))`