WEBVTT
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Use the properties of determinants to evaluate the determinant of this three-by-three matrix.
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The row operation says that when we add a multiple of one row to any other row, the determinant will remain unchanged.
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It’s sensible at this point to label our rows as shown: row one, row two, and row three.
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The first thing we’re going to do is subtract each of the elements from the first row from the elements in the second row.
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This, of course, means that the elements on the first row and the third row remain unchanged.
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To find the first element on the second row, we’ll subtract 20 from 24, that’s four.
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To find the second element in this row, we’ll subtract five from nine, which is also four.
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And to find the third element in this row, we’ll subtract eight from 12, which is once again four.
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Let’s now repeat this process, this time subtracting each of the elements in row three from the elements in row one.
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This time, the elements in the second and third rows remain unchanged.
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To find the first element in this row, we’ll subtract 17 from 20, which is three.
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Then, we’ll subtract two from five, which is once again three.
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And again, we’ll subtract five from eight, which is also three.
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The second property we’re interested in is to do with scaling.
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If we scale a row, the determinant will also be scaled by that same factor.
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If, for example, we multiply a row by three, the determinant will also be multiplied by three.
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We’re going to scale the first row.
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We’re going to divide by three or multiply by one-third.
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Since we need our determinant to remain unchanged, we will need to multiply it by three to cancel this effect out.
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Similarly, we’re also going to scale the second row by dividing by four or multiplying by a quarter.
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We’re going to scale the second row by dividing by four or multiplying by one quarter.
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Once again, we need this determinant to remain unchanged.
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So we’ll need to multiply the entire determinant by four to ensure that this does indeed remain unchanged.
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Multiplying through in that first row by a third, and we get one, one, and one.
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Similarly, multiplying through the second row by a quarter, we also get one, one, one.
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And of course, we said we need to counteract this by multiplying by three and four.
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There is one more fact we can use here.
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The determinant of any matrix 𝐴 is equal to zero, if it has two equal lines.
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We can see that the elements in the first row and the second row are all one.
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They are equal.
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This means that the determinant of this matrix is zero.
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And of course, we need to multiply that by three and four, which is also zero.
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We’ve used the properties of determinants to calculate the determinant to be zero.