Linear Algebra Determinants

Welcome back to educator.com, this is linear algebra, and today we are going to be talking about determinants.0000

Determinants have, determinants are very curious things in mathematics, obviously they play a very big role in linear algebra, but they also play a big role in other areas of mathematics.0007

Now we are not going to be necessarily be dealing with too many of the theoretical aspects, we are going to be more concerned with computation, using them to actually solve problems.0020

But it is good to know that determinants are very deep part of mathematical research, let's go ahead and get started.0031

Okay let us...0039

Take a matrix A, and we will take, we will use our notation A₁₁, A₁₂, A₂₁, A₂₂.0044

Okay, 2 by 2 matrix, we define the determinant of A...0057

... Another symbol for the determinant is a straight line up and do, we will be using that symbol, we will be using both interchangeable A₁₂, A₂₁, A₂₂.0065

So it depends on what it is that you are talking about, you use the lines whether you want to specify the entries.0075

We use this more functional notation, determinant of this matrix A, when we want to simply speak about it in the abstract, so we define it as A₁₁ times A₁₂ - A₁₂ times A₂₁.0083

The pattern is this times that - this times that...0100

... Again a₁₁ along the main diagonal this times that - that times this including the signs.0109

Let's do the same for a 3 by 3 and then we will do some examples, so let's say that B is our 3 by 3.0119

We have B₁₁, B₁₂, B₁₃, B₂₁, B₂₂, B₂₃, B₃₁, B₃₂, B₃₃.0130

Linear algebra is notationally intensive, okay now, the determinant of B is equal to, okay I am going to write it out, and then we will talk about an actual pattern by that we can use.0149

B₁₁ times B₂₂, times B₂₃ + B₁₂ times B₂₃, times B₃₁ + B₁₃ times B₂₁ times B₃₂...0173

... - B₁₁ times B₂₃ times B₃₂ - B₁₂, time B₂₁, times B₃₃...0200

... - B₁₃ B₂₂, B₃₁,0222

Here is the pattern, there are different ways to think about this, and I know that many of you have of ‘course seen determinants before back in high school and perhaps in other areas of mathematics, perhaps in some college courses, in calculus or something like that.0232

Here is the general pattern, notice we have some that are +, +, +, + and some that are -, -, -...0247

... First one is B₁₁, B₂₂, B₃₃, going from top left to bottom right, multiply everything down this way.0261

The second entry, just move over and go down again to the right, B₁₂, times B₂₃, but since you have nothing over here, just go down to this one, because you need three entries, notice each one of these has 3, 3, 3, 3, 3, 3.0271

You need three factors in the multiplication, so it's this times this times this.0289

Now you go to the next one B₁₃, there's nothing here, but you need three of them, so you go here and here, so it's B₁₃, B₂₁ times B₃₂, that takes care of the plus part.0295

Now let's deal with the minus part, go back to the B₁₁, well B₁₁ now go, try going down to the left, well there's nothing here at the left but you need three terms.0308

It's B₁₁, B₂₃, B₃₂, go to the next one over, B₁₂, B₂₁, there's nothing here, but there is one here B₃₃.0317

B₁₃, B₂₂, B₃₁, there are different kind of pattern's that you can come up with, this is simply the best pattern that I personally have come up with to work with 3 by 3.0332

Again, you have probably seen determinants before, so whatever pattern you come up with is fine, I think this works out best, simply because you are going to the right, positive, you are going down the left, negative, if that makes sense.0347

Those have positive signs, when you are moving in this direction, you have negative signs, and of ‘course in this, you have a 3 by 3, each term has to have three things multiplied by each other, okay.0361

Lets do some examples...0372

Let's go back to, actually you know what?, I think I am going to try blue ink, let's define A as (1, 2, 3, 2, 1, 3, 3, 1, 2) okay, so let's do our pattern.0378

Let's see, let's go ahead and put something like that, and we will say the determinant of A, okay, 1 times 1 times 2 is 2, okay + 2 times 3 times 3.0397

2 times 3 is 6, 6 times 3 is 18 + 3 times 2 times 1, 6 okay.0412

Now, - 1 times 3 times 1 is 3, - 2 times 2 times 2, 2 times 2 is 4 + 4 is 8 + this 8 off - 3 times 1 times 3, -9.0423

When we add them all up, hopefully my arithmetic is correct, please check me you should get 6, so again positive this way, negative that way.0445

Let's do a 2 by 2, let's say B is equal to (4, -7, 2, -3) so now we have some negative entries, okay, the determinant of B is equal to this times this.0460

4 times -3 is -12 - this times this, this times this is -14, a - sign has to stay, so it's -12 - (-14) - 12 + 14, it is equal to 2.0479

Okay, so these signs here, this + here, + here, + here, -, -, -, they always stay there, it doesn't matter what these numbers are,0500

If this is negative, then a negative times a negative is positive, but you have to have three positive terms in a 3 by 3, you have to have 3 negative terms in a 3 by 3.0512

Negative in this case doesn't depends on what these numbers are, but those negative signs and these positive signs have to be there.0522

They are not part of the arithmetic; they are part of the definition of the determinant, okay.0529

Let's see here...0538

let's go over some properties of the determinants, just like we did properties of matrices, we will talk about some properties of determinants, so let A be an N by N matrix okay, then the determinant of the A transpose is the determinant of A.0543

In other words if I take A, take the transpose of it, and if I, then i take the determinant, it's the same as the determinant of A, no change, if you have a matrix and you interchange two rows or two columns this way or this way of the matrix, the determinant changes sign, so it goes from positive to negative, negative to positive.0559

Positive to negative, negative to positive, if two rows or columns of a matrix are equal, then the determinant equals 0, that isn't simple.0579

If a row or a column of A is entirely 0's then the determinant again is equal to 0, if a single row or a column is multiplied by a non-zero constant R, non-zero, then the determinant is multiplied by R.0593

The whole determinant is multiplied by R, if one row or column is just multiplied by r, let's do an example...0609

... We will let A = (1, 2, 3(1, 5, 3) and (2, 8, 6) okay, we want to find the determinant of A, in this case i am actually going to write it with this symbol, and you will see why in a minute.0621

I am going to rewrite it (1, 2, 3, 1, 5, 3, 2, 8, 6), in this case, using some of the properties particularly the one where we say if we multiply by a particular constant, I am going to use something that's going to be a Kent 2, factoring out, that you are used to from algebra, that's why I used this.0641

So I want you to see all of the entries, that's equal to... So notice this is (2, 8, 6).0663

You can divide this by 2, which means I can actually factor our a 2 from here, so i am going to put a 2, and i leave the other rows the same(1, 5, 3) and this becomes (1, 4, 3).0672

I can factor out a 3 here too, so I have 2, times 3,(1, 2, 1), I can factor out a 3 from this column, (1, 5, 1, 1, 4, 1) okay.0686

Now notice, I have a column of 1's and another column of 1's, two column that are the same, so now the determinant is equal to 2 times 3.0705

And when I have something, the two columns are the same, the determinant is 0, so saved myself a lot of problems, I didn't have to go through that whole strange, this diagonal, that diagonal, this entry, that entry, positive negative.0716

I used the properties of determinants, to actually make my life a lot easier, and I was able to find the determinants of the 3 by 3 pretty quickly, just by some standard algebraic manipulation, okay...0731

... Property number 6, if a multiple of one row or column of A, is added to another row or column, then the determinant is unchanged, remember that process of elimination that we did, we are doing Gauss Jordan reduction, Gauss Jordan elimination, where we multiplied some multiple of one row and added it to another.0750

When you do that, you are creating an equivalent system you remember, so the determinant doesn't change because the system is equivalent, now the seventh property, very important, if A is upper triangular, that means all entries below the main diagonal are 0, then the determinant a, of A is the product of the entries on the main diagonal.0771

A, upper triangular matrix, looks like let’s just say (1, 2, 3) let's say (3, 4, 6)...0792

... everything, so this is the main diagonal, okay, everything below the main diagonal is 0, so notice (0, 0, 0) but there are entries on the main diagonal, some of them can be 0's but they are not all 0's, okay.0807

Upper triangular means the upper part, the upper right hand art is the shape of a triangle, okay, when that's the case, then the determinant is just that.0820

It's kind of nice, so if you can actually turn it into buy a bunch of manipulation that you do to matrices, our elimination, interchanging rows, multiplying, you know one row by another, adding it to another row.0831

If you can actually change it to an upper triangular matrix and you just multiply those entries and you have your determinant, let's do an example, okay...0843

I have, I will go ahead and put my determinant symbol right there already (4, 3, 2) (3, -2, 5) and (2, 4, 6), okay i am going to go ahead and factor out a 2 from the third row right here.0858

That's equal to 2 times (4, 3, 2) (3, -2, 5), (1, 2, 3), okay.0879

I am actually going to switch this row, the third row and the first row, I am just going to switch them, and when I do that, I change the sign of the determinant, so I just take a -2 in front of that, and then I go (1, 2, 3), I will lead the second row the same (2, 5, 4, 3, 2) and these are pretty simple things that we are doing here.0892

Now, I have a 1 here and i have a 3 and then the 4, I am going to multiply this first row by -3 added to this, okay...0916

We end up with -2 times (1, 2, 3), (0, -8, -4) (4, 3, 2), now I am going to do the same thing with this row right here.0928

I am going to multiply this first row by -4, added to the third row, so I end up with -2 times (1, 2, 3), (0, -8, -4), (0, -5, -10), okay.0943

I am going to factor up, notice here, 4, there is an 8, there is a 4, I am going to go ahead and factor out a 4, so -2, we will take a 4 and I am going to also take out a 5 here, okay.0963

5, that turns it into (1, 2, 3)...0979

... (0, -2, -1) and (0, -1) oops, that is a -10, if I divide by 5, which should give me a -2.0986

Okay, so all I have done is factor up...1001

... I am actually going to do one more thing here, I am going to switch these rows simply because I want the 1 on top of the 2, personal choice, I don't necessarily need to do it, so again when I switch a row, I change the sign, so I get rid of that negative sign 2 times 4 times 5.1007

(1, 2, 3), (0, -1, -2), (0, -2, -1) okay, now I am going to multiply this by positive 2, this second row by +2 added to this one to get rid of this -2.1026

Okay, and when I do that, I get 2 times 4 times 5 (1, 2, 3), (0, -1, -2), (0, 0, 3), now have a...1045

... Upper triangular matrix, 0's entries along the main diagonal, now my determinant is equal to, 2...1063

... I am going to skip the parenthesis, times 4, times 5, times 1, times -1, times 3.1074

just this times, this times that, and now the determinant of this is just the entries along the main diagonal, and I have just a straight multiplication problem, and I should end up with -120, so properties, I have take a matrix...1086

... Subjected it to a bunch of, you know properties, simplified a little bit in order to find the determinant, okay...1102

... Few more properties to go, number 6 is if I take two matrices and I multiply them and then take the determinant, well it's the same as just taking the determinant of the first one times, the determinant of the second one, so determinant of A times B is equal to the determinant of A times determinant of B, reasonably straight forward there.1116

Now, if A is nonsingular, if it's invertible, if I have the inverse and if I take the determinant of it, it's the same as taking 1 over the determinant of the original matrix, notice this is not...1135

... Again we are not talking about 1 over A, 1 over A, A is a matrix, division by a matrix is not defined, but the determinant is a number, so division by a number is defined.1150

Let's go ahead and do an example, we will let A = (1, 2, 3, 4), we know that the determinant of A = 1 times 4, just 4 - 2 times 3, 6 = -2, okay.1162

Now, when we use our math's software to calculate the inverse of this matrix, we get the following (-2, 1, 3 halves, -1 half), when we take the determinant of the inverse, we get -2 times -1 half.1186

-2, I will write this one out, times -1 half, -...1210

... 1 times 3 halves, -2 times -1 half is a 1, -3 halves = -1 half.1219

Well, we said that the determinant of the inverse, of ‘course, we said that it's equal to, we want to confirm that, 1 divided by the determinant of the original.1233

Well the determinant of the inverse is -1 half, is it equal to 1 over -2, yes...1248

... If you want to know the determinant of the inverse, instead of finding the inverse and getting the determinant, you can just find the determinant of A and take the reciprocal of it.1260

Again, this is not 1 over the matrix A, this 1 over the determinant of A, the determinant is a number, the matrix itself is not a number.1267

And that covers determinants, thank you for joining us at educator.com, linear algebra; we will see you next time, bye, bye.1280