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 given three-by-three matrix. We can do this by expanding over
any of the rows or columns. And this would work. We can use this to get the correct
answer. However, we can notice something
interesting about this matrix. We can see that the third row
contains only zeros.
Therefore, if we were to expand
over the third row, every term in our expansion would have a factor of zero, which
means the determinant would evaluate to give us zero. In fact, this result holds true in
general. If we have a square matrix where
any of the rows or columns only contains zero, then we can conclude the determinant
of that matrix must be equal to zero. And in particular, 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
only containing zeros.
