Invertible and noninvertibles matrices

Let me just write a general two by two matrix A. So let's just say its elements are A, B, C, and D. Now, from previous videos, we have learned how to find the inverse of our matrix A. The formula that we went over, the inverse of our matrix A, is going to be equal to 1 over the determinant of our matrix A times what is often called the adjoint of our matrix A.

That's you are swapping the A and the D, so you have A, D here and an A here, and you're making the B and the C negative. This works for a two by two. When you do it, when you go to the three by three case, it gets a little bit more complicated. So we're going to have a negative B here and a negative C here.

Or another way to think about it, the determinant of a 2 by 2 matrix is going to be 1 over A, D minus B, C. A, D minus B, C, and then you multiply that times the adjoint. So that is a fancy word, that's a fancy word for it. So we have D, negative B, negative C, and A.

So that is all review. If it's not, I encourage you to review it in other places on Khan Academy. But what we're going to ask ourselves in this video is: is there a situation where a matrix is not invertible? So pause this video and see if you can figure out a scenario or a general set of scenarios where a matrix is not going to be invertible.

One way to think about a matrix that is not invertible is that for that matrix, we would somehow not be able to make this calculation. Now, no matter what A, B, C, and D are, we can always find an adjoint matrix. But what's a situation where 1 over the determinant is— we're not going to be able to calculate it?

Well, a situation where the determinant— the determinant of our matrix A is equal to zero. Or I could write the determinant of our matrix A is going to be equal to zero. In that situation, we're not going to be able to calculate 1 over the determinant because that is undefined.

And so we say, we say A is not invertible if and only if— I’ll do this two-way arrow right over here— our determinant for our matrix A is equal to zero. Because if our determinant is equal to zero, we're going to have to divide by zero when we find the inverse.

Now, let's explore what that would look like a little bit more. If our determinant is going to be equal to 0, that means that this expression right over here is going to be equal to 0. So that means that A, D minus B, C is equal to 0. We can add B, C to both sides; that means that A times D is equal to B times C.

And then we can algebraically manipulate this a little bit more. If we divided both sides by B and D, so let me just do that. If I divide both sides by B and D, then the D's cancel out here and the B's cancel out here. So this could be A over B is equal to C over D.

So we have a situation where you could think of it as the ratio between A and B is equivalent to the ratio between C and D. You could also think about it the other way: you could divide— let's start with that original equation A, D is equal to B, C. If you divided both sides by C and D; so you divide both sides by C, D, what you're going to be left with is that A over C is going to be equal to B over D.

And so the ratio between A and C is equal to the ratio between B and D. So all of those situations, you are going to have a non-invertible matrix because the determinant is going to be equal to 0.

And so let's look at a few examples. I want you to tell me whether the following matrices are invertible or not. So let's say I have the matrix 5, 1, 3, 2. Is this an invertible matrix? Pause this video and try to figure it out.

So the determinant of this matrix is going to be equal to 5 times 2, 10 minus 3 times 1, minus 3, which is equal to 7. And since this is not equal to 0, we know that this matrix is invertible. We can find an inverse.

Now, what about this matrix? So 10, 20, 250, and 500. Is this matrix invertible? The determinant here of 10, 20, 250, and 500, this is going to be equal to 10 times 500, which is 5000, minus 20 times 250, which is 5000, which is equal to zero.

And once again, if you wanted to find its inverse, you would take your adjoint matrix and multiply it times one over the determinant. But one over zero is not defined, so this is not invertible.