College Algebra: Inverses: 2x2 Matrices

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So now the question is how do you actually find the inverse of a matrix. And by the way, the only matrices that you can find inverses of, if they exist at all, are square matrices. And in particular only those square matrices that are nonsingular, that means the determinant of those matrices is not 0. It’s actually like you can’t divide by 0. So how do you find the multiplicative inverse? Well let’s just start off with a two by two case, two by two matrices. I’ll first show you a little formula for it, and then I’ll show you how to think about the formula.
So here’s the formula. Suppose we have A equals, and let’s just say it’s a two by two matrix, a general matrix. So I have a, b, these are the entries, c and d. Then the question is what’s the inverse of this matrix? Well first of all, let me show you the notation. The notation we use to show the inverse matrix is the following. We write, just like with numbers, we write A with a -1 exponent. And that’s just read A inverse. And what’s the definition of A inverse? Well it starts off with a scalar, we take 1 over the determinant of A. So you have to compute the determinant of A. That’s just that number ad - bc. That’s a number. And you take its reciprocal. Okay, and then you scalar multiply that by the following new matrix. It’s still going to be a two by two, of course, and what’s the recipe? The recipe is you take this diagonal here, and you switch them. So I just switch the roles of these things. So I put a d here, and then I put an a here. And then here actually I don’t do anything but just replace the signs by negatives. So I just switch the signs of these people. And these people all I do is actually flip these numbers. So I flipped here, and just put negative signs in front of those people there. That should give me the inverse.
Let’s try it with a particular example. So suppose I take a look at the following. Suppose that A were to be the matrix 1, 3, -1, 5. And the question is what’s the inverse? Well the inverse would be, what do I do? Well first I have to compute the determinant. So I see a 1 x 5, which is 5 - 3 times -1, so that’s -3. So 5 minus -3 is actually just 5 + 3, which is 8. So I take the reciprocal of that, so I write . And then I multiply that by what matrix? Well I flip these people so I have a 5 and a 1 here. And then I put negative signs in front of these people. So I have a -3, and then I have a negative -1, which becomes a positive 1. And remember what you do when you have a scalar multiple times a matrix, what you see is just times each and every term. So I see . I see . I see , and I see . I just took every single term and multiplied it through by . And that should equal the inverse. Let’s check it.
How would I check it? The way I check would be just to say, okay, well this product in fact should be the identity. Well let’s see what that product is. So if we take that product, I see 1 x , which is . Then I add to it . Well + is just , which is 1. Well that’s a good way to start, because that’s the first element of the identity. Now this should be a 0 here. Let’s see if that’s true. Here I see a , and then I have a + . That actually combines to give 0. Neat, this should also be a 0. Let’s see. Here I see a - and . That combines to give zero. And this last term should be a 1. Let’s see. We have a minus . That’s just + , and + , well + = 1. So look. This actually checks. I get the identity back. So in fact, this really is the inverse event.
So finding inverses of two by two matrices actually is not that big of a deal. All you do is you divide by the determinant, and then you just switch the roles of these two people on this diagonal, and then these two people on that diagonal, you put negative signs in but keep them in place. That’s all there is to it. Up next we’ll take a look at the slightly more exotic situation when you have a three by three matrix that you want to find the multiplicative inverse for. It gets a little tricky, but first let’s just revel in the celebration of finally getting this right. I’ll see you soon.