 We define the determinant of a square matrix to be a function that satisfies a set of properties. So remember, definitions are the whole of mathematics, all else is commentary. This definition alone allows us to compute the determinant of any matrix. So let's say I want to compute the determinant of this 3x3 matrix. So our linearity property allows us to break the determinant of the matrix into a sum of the determinants of 3 matrices, where we see that if we take the first rows of these 3 matrices and add them together, we actually get the first row of the matrix we're interested in. And so I can find the determinant of this matrix by finding the determinant of these matrices. Except now we have to find 3 determinants. So maybe we should look at simpler matrices first and see if we can say anything about the determinant in general. So let's think about that. We do know how to compute the determinant of a 2x2 matrix and since the determinant of a 2x2 matrix relies on multiplying the entries, let's start by making the entries as simple as possible. For multiplication, simple means zero. And so we might ask, what if a matrix has a row or column of zeros? So let's find the determinant of this matrix where we have a column of zeros. Again, every problem in linear algebra begins with a system of linear equations, so let's let the determinant be the unknown that we're looking for. So again, definitions are the whole of mathematics, all else is commentary. The definition of determinant says that the determinant is linear in the entries of a row or column. Well remember, if t is a linear transformation, the transformation applied to a scalar multiple is going to be c times the transformation applied to the vector. Now it's important to recognize that the linearity applies to the entries of a row or a column. So the scalar multiple should always be viewed as the scalar multiple of a single row or a single column. Suppose I take c not equal to zero. So I'll take my matrix and I'll multiply one row or column, how about the middle column, by our constant c. Linearity says that I'm going to get c times the determinant. But let's do a little simplification. This second column over on the left is all c times zero, so it's still zero, and on the right I have the same matrix that I started with. So remember, x was the value of the determinant, and so that gives me the equation x equals cx, and so I'll solve for x, and I have my determinant. And this suggests the following property. Let m be a square matrix. If m has a row or column of zeros, the determinant of m is equal to zero. By a similar set of arguments, the linearity property again gives us the following. Let b be the result of multiplying any single row or column of a by a constant c. Then the determinant of b is c times the determinant of a. And we can show this by, well, this is a homework problem, so you should do it. So what about the determinant of a diagonal matrix? Well, let's summarize everything we know. First, definitions are the whole of mathematics, all else is commentary, and so let's pull in that definition of determinant. Now, just from the definition of the determinant, we were able to show that if we have a row or column of zeros, the determinant will be zero. And so we don't have a row or column of zeros, so this doesn't help us. Also, just from the definition, we were able to determine that if b was the result of multiplying any single row or column of a by a constant c, then the determinant of b is going to be c times the determinant of a. Well, let's see if we can use that. So the first thing we might notice is the first row of the matrix is really 2 times 1, 0, 0. And so by linearity, we can rewrite this determinant as, now that might not seem very useful because then we still have to compute the determinant of this matrix. But wait, this second row is the result of multiplying 3 by 0, 1, 0. And so we can write down this relationship. And lather, rinse, repeat. The third row is the result of multiplying 5 by vector 0, 0, 1. So we can write down this relationship. And, well, that doesn't really help us because we still have this determinant of a matrix we have to figure out. If only there was some way of knowing the determinant of this matrix. Oh wait, definitions are the whole of mathematics. All else is commentary. The definition of the determinant tells us that the determinant of the identity matrix is 1. So we do know the determinant of this matrix, which means we also know the determinant of the given matrix. And we can generalize this result to prove that the determinant of a diagonal matrix is the product of the entries on the diagonal. And in fact, we can go further. The determinant of an upper or lower triangular matrix is also the product of the entries along the main diagonal. And now we're ready to do an important example. Let's consider this matrix. Now our linearity property allows us to find the determinant of this matrix as the sum of the determinants of three matrices. And again, we've taken our first row and broken it apart into 2, 0, 0, 0, 7, 0, 0, 0, 5. And importantly, everything else stays the same. And notice these last two matrices have a column of 0s. And our theorem says we know what that determinant will be. So the determinant of our original matrix is the same as the determinant of a much simpler matrix. Well, that was useful. Let's do that again. And so our determinant of the simpler matrix, I can use linearity and rewrite this as the sum of the determinants of two matrices, where this time I'm going to break apart that second row into 0, 3, 0, and 0, 0, 1. Now notice this first matrix is a lower triangular matrix, and we know what to do with the determinant of a lower triangular matrix. This second matrix isn't a lower triangular matrix, but if we switch the second and third rows, it becomes a triangular matrix. And if we switch adjacent rows, that's going to change the sign of the determinant. So that means we'll be subtracting the determinant of this matrix. And so our determinant is going to be the difference between the determinant of two triangular matrices, which we can compute, and we get our determinant.