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