Find the inverse of the following matrix. 𝐴 is equal to negative one, one, negative one, eight.
Remember, for a two-by-two matrix 𝐴 which is equal to 𝑎, 𝑏, 𝑐, 𝑑, its inverse is found by multiplying one over the determinant of 𝐴 by 𝑑, negative 𝑏, negative 𝑐, 𝑎.
A good way to remember this is that we swap the elements in the top left with the bottom right. And then we multiply the elements in the top right and bottom left by negative one, essentially changing their sign. The formula for the determinant of that same matrix is 𝑎𝑑 minus 𝑏𝑐. We find the product of the top left and bottom right elements and then subtract the product of the top right and bottom left.
Notice this means if the determinant of 𝐴 is zero, the inverse cannot exist, since one divided by the determinant of 𝐴 would be one divided by zero, which we know to be undefined.
It’s always sensible then to begin by calculating the value of the determinant of the matrix. We multiply the top left and bottom right elements. That’s negative one multiplied by eight. And then we subtract the product of the top right and bottom left. That’s one multiplied by negative one. Negative one multiplied by eight is negative eight. And one multiplied by negative one is negative one. That’s negative eight plus one, which we know is negative seven.
Now that we know the value of the determinant, let’s substitute that into our formula for its inverse. One over the determinant is one over negative seven or negative one-seventh. We then switch the elements in the top left and bottom right. We swap negative one with eight.
Finally, we multiply the element in the top right and bottom left by negative one. That’s essentially changing their sign. And we get negative one and one. Now we could at this point multiply each of the individual elements inside this matrix by one-seventh. But that doesn’t really simplify it.
What we can do, though, is multiply through by negative one. And in doing so, we see that the inverse of the matrix 𝐴 is a seventh multiplied by negative eight, one, negative one, one.