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.