Professor Dave again, let's talk determinants. We now know how to add, subtract, and multiply matrices. So it's time to learn one more thing we can do. We need to be able to find the determinant of any square matrix. Let's recall that a square matrix is any matrix with an equivalent number of rows and columns, whether two or ten.
So let's learn this algorithm now. Starting with the simplest square matrix, a two by two matrix, with entries A, B, C, and D, the determinant, represented by these vertical lines on either sides of the entries, will be a scalar with the value of AD minus BC. In other words, we find the product of the upper left and lower right entries, and subtract from that the product of the upper right and lower left entries.
So for example, a two by two matrix with entries two, one, negative six, and four will have a determinant of two times four minus one times negative six. This is eight minus negative six, or fourteen. This seems simple enough, and in fact, that's all there is to it when finding the determinant of a two by two matrix. However, we did say that we can find the determinant of a square matrix of any size, so let's see what happens when looking at a three by three matrix. Say we have the following matrix, with entries A1, A2, A3, B1, B2, B3, and C1, C2, C3.
Finding the determinant of this will already be much more complicated than for a two by two matrix, because its determinant must be broken down into a series of second order determinants. We will take A1 and multiply it by the determinant of a new two by two matrix comprised of B2, B3, C2, and C3. So it's like we ignored the row and column that A1 is a part of, and kept the rest for a new matrix.
From this we subtract A2. times the determinant of another two by two matrix, this time ignoring the row and column that A two is a part of, leaving us with B one, B three, C one, C three. And to that, we add A three times the determinant of one more two by two matrix, containing B one, B two, C one, C two.
Pay close attention to the fact that we used a minus sign here, and then a plus sign here, this rule must be followed. From here we just evaluate these three simple determinants using the method we already know for two by two matrices. That would give us some products to evaluate and we simplify until we have a final scalar value. Let's practice this with an example. Here we see the entries one two negative one three zero one negative five four two.
If we want to find the determinant of this three by three matrix, once again all we do is take this first entry, one, and multiply by the determinant of this two by two matrix. From that we subtract the product of the second entry from the first row, two, and the determinant of this two by two matrix. And to that we add the product of the third entry from the first row, negative one, and the determinant of this last two by two matrix.
Now we just have three simple determinants to calculate. The first is zero times two minus one times four, or negative four. The second is three times two minus one times negative five, or 11. And the third is three times four minus zero times negative five, or 12. We multiply each determinant by the adjacent constant, then we perform the addition and subtraction, and we should get negative thirty-eight as our answer.
Now how would we find the determinant of larger matrices still? As the matrices get bigger, the algorithm doesn't really get any harder, we just end up adding many more steps very quickly. For a four by four matrix, We will take the first entry and multiply by the determinant of this 3x3 matrix, minus this second entry times the determinant of this 3x3 matrix, plus this third entry times the determinant of this 3x3 matrix, minus this fourth entry times the determinant of this 3x3 matrix.
So we are doing the same thing we did before, going across the first row and blocking out the row and column containing a particular entry, and multiplying by the relevant determinant. Also notice that we subtract a term, then add the next, then subtract the next. This would continue alternating for as many terms as were necessary.
Now that we are here, we have four separate three by three matrices, we need to find determinants for, and as we just learned, each of these will involve the determinants of three separate two by two matrices. So we have definitely made quite a bit of work for ourselves, but as long as you remain organized and stay on top of your arithmetic, there is no reason that you can't find determinants for matrices of this size or even larger. Now that we know how to find the determinant of a matrix, let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so I can keep making content, and as always feel free to email me professordaveexplains at gmail.com