00:01
Consider the following elementary row operations and a 3x 3 matrix a.
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The first row operation applied to a is negative to row 1 plus row 2 is assigned to row 2 and that give us the matrix e1 times a where e1 is an elementary matrix.
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And to that matrix we apply the row operation for row 1 plus row 3 assigned to row 3 and we give that give us another matrix obtained by multiplying to the left by e2, the previous result e1 times a.
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Then we apply a third operation and we get the next matrix and final fourth row operation to get matrix b, which is e4 times e3 times e2 times e1 times a in this is equal to this matrix here.
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We want to find the determinant of a, calculate or find the elementary matrices e 1, e3, and e4, and find a matrix a, finally.
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So there are two things, main things to know in order to solve this problem.
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The first thing is that we know that mathematically is the same.
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Applying this role operation, or any role operation like this to matrix a, is the same as multiply matrix a to the left by an elementary matrix, in this case e1, which is a from the identity matrix by applying on that identity matrix the same operation.
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That means that we can calculate easily e1 simply by applying the operation indicated here to the identity matrix 3x 3.
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The same for any of the other role operations.
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That is e2 here is obtained from the identity matrix by applying to to that identity matrix is through operation here.
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Same thing for e3 and for e4.
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So we are going to calculate first the elementary matrices.
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That is we are going to solve part b first.
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So in part b, let's see the identity matrix 3 by 3 is 1 -0 -0 -0 -0 -0 -0 -1.
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And we are going to apply this elementary row here, elemental row operation to identity matrix 3 by 3.
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We get put here the elementary operation negative 2 row 1 plus row 2, sign to row 2.
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That is i'm talking about this elementary row operation here.
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So now we apply that to this identity matrix and what we get here.
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We're going to modify the second row so the first and the third remain the same and the second will be what it has now here minus two times the first row so zero minus two times one is zero minus two is negative two okay then one minus two times zero and zero minus two than zero is zero so we get this and this is e1 and the the key thing is that mathematically is exactly the same applying the row operation to matrix a, then multiply matrix a to the left by these elementary matrix.
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Of course, when we are doing this kind of triangular reduction of matrix b by elementary row operations, we don't do the multiplications here.
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This is more of a theoretical thing that we use for some results.
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But in the practical way, we apply the raw operation directly to matrix a, and we are going to obtain each partial matrix up to the last one where we have an upper triangular matrix.
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So, but this is, in this case, we are using the fact that we can calculate matrix e1 simply by knowing the raw operation.
04:55
We apply that role operation to identity matrix and we get e1.
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So let's do the same thing to calculate e2.
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And so let's see to these 3x3 identity matrix, we apply the second row operation here.
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I'm going to write that row operation here to know what it is.
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4 row 1, it's row 3, assigned to row 3.
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Good so let's see what we get here so we are going to modify row three so row 1 and 2 are the same and so we get so row 3 will be what it has right now here plus 4 times the first row so 0 plus 4 times 1 is 4 plus 4 times 0 and 1 plus 4 times 0 is 1 so this is matrix e2 in fact.
06:31
So we have found e1e2.
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We can find now e3.
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So we start again with the 3 by 3 identity matrix here.
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And we apply the row operation that comes now.
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This one here is one half row 2 goes to rows 2.
06:56
One half row 2 goes to row 2.
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And we get one half of row 2 simply.
07:04
So the first is the same, the third is the same, but the second is half.
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All vows in the second row are divided by 2, so we get 0, 1 1 1ā2.
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And that is e3.
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And finally to get e4, we start with the 3x3 identity matrix.
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And we apply the row operation here.
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So i'm going to write it here again.
07:45
Is negative row 2 plus row 3 goes to row 3 means that first and second row rows are the same and so row 3 minus row 2 but basically so this row here is 0 minus 0 is 0 is 0 minus 1 and 1 minus 0 is 1 so we get that matrix and then check something here.
08:34
So let me see this is e4 and this is here e3.
08:45
Okay, so we have calculated in part b, the elementary matrices e1, e2, e3 and e4.
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Good.
08:58
So now knowing that we can calculate the determinant of a, part a.
09:06
So we know that b is the matrix 3, 2, negative 1, 0 ,1, 3, and 0 ,0, negative 4.
09:26
This is equal, we know to e4, that is b3, b is the last matrix after applying the four row operations.
09:36
It means it has a form e4 times e3 times e2 times e1 times 8 and so now here we know this matrix is here is upper triangular okay so when a matrix is upper triangular we know how to calculate its determinant by simple multiply the the elements the values in the main diagonal.
10:10
So b, let me put that here, b is upper triangular, and that means that the determinant of b is equal to 3 times 1 times 94, that is product of entries of b on main diagonal.
10:59
And that's a property that is true, sorry, that is true for.
11:04
Any upper triangular, in fact lower triangular also a diagonal matrix.
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Okay, so we have that.
11:13
So the determinant of v is ridley known is netting 12.
11:20
Okay, now we apply another property of the determinant we know is the determinant of the product of matrices is the product of the determinants of the factors...