Calculating the inverse of a matrix

But wait – there's more! We can use the determinant of a matrix to calculate its inverse. Here is the formula to calculate the inverse of a 2 × 2 matrix:

The first thing to notice is that the reciprocal of the determinant is used. Now, what if the determinant is zero? We can't divide by zero! So the formula is undefined, but remember that if the determinant is zero, it doesn't matter because the matrix is not invertible to begin with!

The other thing to notice is that a and d just switch position. Then, b and c are just multiplied by -1.

That was a lot of words! Let's look at an example with the following matrix:

Let's calculate the determinant first:

Alright, let's use that to calculate the inverse:

And there you have it. Let's make sure that it is the inverse by using the definition we...