 So if you've been working in linear algebra, there is a particular expression that keeps reoccurring. So, for example, the system of linear equations with a coefficient matrix has a solution, or not, depending on the value of what we might call the difference of the cross products, A D minus B C. Meanwhile, if we consider the transformation corresponding to the same matrix, we find that this would change the area and orientation of a unit square according to the difference of the cross products, A D minus B C. And when we found the inverse of a matrix, we found that the inverse of the matrix relied on dividing by the difference of the cross products, A D minus B C. And that says this difference of the cross product seems to be an important quantity, and so we'll define the determinant of a 2 by 2 matrix as follows. Let A be this matrix. The determinant of A, written between bars, is the difference of the cross products, A D minus B C. It's important to remember that there are only so many symbols, and so these vertical bars serve multiple purposes in mathematics. These, especially here, do not mean absolute value. The determinant could be positive or it could be negative. So what are the properties of the 2 by 2 determinant? We claim the following. First, the determinant of the 2 by 2 identity matrix is 1. Next, switching two rows or two columns of A will change the sign of the determinant. Next, if a matrix A prime is formed by multiplying a row or column of A by k, then the determinant of A prime is k times the determinant of A. And finally, the determinant is a linear operator on the entries of each row and each column. In other words, if I have the determinant of this matrix, I can break that first row into A0 and 0B. And if we keep the other entries the same, then the determinant of the original will be the sum of the determinant of the two other matrices. And since this is on the Internet, you know it must be true because I have my own YouTube channel. But you should prove these statements anyway. And since all matrices are 2 by 2 matrices, we can stop here. Well, since linear algebra deals with matrices of any size, we should try to generalize the concept of the determinant. So let's think about that. Could we have a determinant for an n by m matrix with n not equal to m? Well, possibly, but remember the uses of the determinant. It told us whether a system of equations had a unique solution, but if n is not equal to m, then the system of equations might have no solutions or an infinite number, but it won't have a unique solution. The determinant also told us whether a matrix had an inverse. But if n is not equal to m, the matrix won't have an inverse. And finally, it told us the area of a unit square under a transformation. But if n is not equal to m, the transformation takes an m-dimensional object and produces an n-dimensional object, so the area no longer exists. We don't talk about the area of a cube. And so in some sense, we already know the answer to these questions. And since we already know the answers to these questions if n is not equal to m, there's no reason to define the determinant for n not equal to m. And so that means determinant only makes sense for square matrices. And so this leads to our definition. What we want to do is to preserve as many of the nice results of our 2x2 determinant. So let m be a square matrix. The determinant of m, written this way, is a function that satisfies the following properties. Now, the determinant of a 2x2 identity matrix is 1, so we'll make that one of our defining properties of the determinant. The determinant of any identity matrix is 1. Switching two rows or two columns of a will change the sign of the determinant. Now, since a is a 2x2 matrix, this also means that the rows or columns that we're switching are adjacent. So that means if we want to switch rows, they should be adjacent rows. And so our definition requires that if m' is the result of switching adjacent rows or adjacent columns of m, the determinant of m' is minus 1 times the determinant of m. For a 2x2 matrix, if we multiplied a row or column of the matrix by k, then the determinant would also be multiplied by k. And so we'll require that if m' is the result of multiplying a row or column of m by a scalar k, the determinant will also be multiplied by k. And finally, we have this beautiful linearity property. The determinant is a linear operator on the entries of each row and each column. And so we'll require that the determinant is linear in the entries of a row or a column. Now, it turns out that if we maintain this requirement that the determinant is linear in the entries of a row or column, we actually get the scalar multiplication property. And this is actually an example of the power of linearity. If we assume that the determinant is linear in the entries of a row or column, we don't actually need to require the scalar multiplication property. We can actually prove it. So we'll drop that scalar multiplication property and leave this as our definition of the determinant. The determinant of the identity is 1. Switching adjacent rows or columns changes the sign. And the determinant is linear in the entries of a row or column. And at this point, well, definitions are the whole of mathematics, all else is commentary. We know how to calculate the determinant of any matrix. Well, more or less, there is a lot of commentary to go. So let's start off with a simple matrix. Now, whatever the determinant is, it must satisfy the definition. So let's pull the definition in again. So our definition tells us the determinant of the identity matrix is 1. And so the determinant of this matrix is not the identity matrix. Well, that's not quite as useful. But we do know that if we switch adjacent rows or adjacent columns, the sign of the determinant will change. And so we might notice the first two rows are the same. So remember, every problem in linear algebra begins with a system of linear equations. In this case, since we're trying to find the determinant, we're trying to solve a system of equations where the determinant is one of the unknowns. So we'll let our determinant be x. So the definition of the determinant tells us what happens if we switch adjacent rows or adjacent columns. So let's consider our matrix and let's switch the first two rows. OK, ready? Ready? OK, here it is. And since the first two rows were the same, switching them doesn't actually change the matrix. However, it does change the determinant. And so whatever the determinant of the original matrix is, the determinant of the matrix obtained when we switch the first two rows is minus one times that determinant. And so that gives us our linear equation, x equals minus x. And we'll solve, and we find that x is equal to zero. And so the determinant of this matrix where the first two rows are the same is going to be zero. This can be generalized and leads to an important result. Let m be a square matrix. If two rows or two columns are the same, the determinant of m is equal to zero. But what if the rows and columns are different? We'll take a look at that next time.