Question Video: Solving Determinants Using Properties Mathematics

Find the value of [4, 1 −8 and −6, 3, 6 and 0, 0, 0] .

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Video Transcript

Find the value of the determinant of the three-by-three matrix four, one, negative eight, negative six, three, six, zero, zero, zero.

In this question, we’re asked to evaluate the determinant of a three-by-three matrix. And we might be tempted to jump in straightaway and evaluate this determinant by expanding over the first row. And this would work; we would get the correct answer. However, whenever we’re asked to evaluate a determinant, we can always check to see if we can simplify this problem by using properties of determinants. And in this case, we can notice the third row of this matrix is all zeros.

We can then recall the following fact about the determinants of matrices. If all of the entries of a row or column of a square matrix is zero, then its determinant will also be equal to zero. And it’s worth pointing out, in this case, this is exactly the same as saying that we’re going to calculate the determinant of our matrix by expanding over its third row, since then in our calculation of the determinant every single term would have a factor of zero. So, the determinant of this matrix would still be zero. In either case, we were able to show the determinant of the three-by-three matrix four, one, negative eight, negative six, three, six, zero, zero, zero is equal to zero because it has a row entirely made of zeros.

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