Video Transcript
Evaluate the determinant of the
three-by-three matrix one, negative nine, negative six, negative eight, four, one,
two, negative one, nine.
Now when we’re looking to evaluate
a determinant of a three-by-three matrix, then what we do if we take a look at the
first row, and we use the first row as coefficients. And what we do is we multiply in
turn each of these by the submatrix, so the two-by-two submatrix that’s formed when
you delete the row and column that that value is in.
It’s also worth noting that our
coefficients have to follow a pattern. So, our first column is positive,
so the coefficient is multiplied by positive one. The second column is negative, so
multiplied by negative one. Third column, positive. So, we’re gonna use that in a
second when we put it altogether.
So, the first coefficient we’re
looking at is the top-left term. So, it’s one because it’s in the
first row. And we’re gonna multiply this by
the determinant of this submatrix, which is formed, so the two-by-two submatrix when
we delete the row and column that the one is in. So, it’s gonna give us one
multiplied by the determinant of the two-by-two submatrix four, one, negative one,
nine.
Then, next, we’re gonna have minus
negative nine. And that’s cause, as we said, the
second column has to be negative. And we’ve already got negative nine
as our coefficient. Then, multiplied by the two-by-two
submatrix, again, formed when you delete the row and column that the negative nine
is in. So, it’s the determinant of the
submatrix negative eight, one, two, nine. Then, finally, we’re gonna have
minus six multiplied by the determinant of the two-by-two submatrix negative eight,
four, two, negative one. And it stayed as negative six as
the coefficient because, as we said, this third column is positive. So, the sign stays the same.
So, now the next stage is to know
how to deal with the two-by-two determinant. Well, to find the two-by-two
determinant, what we do is if we’ve got the two-by-two determinant of the matrix 𝑎,
𝑏, 𝑐, 𝑑, you multiply 𝑎 by 𝑑, so we diagonally multiply, and then subtract 𝑏
multiplied by 𝑐. So, first of all, we’re gonna have
one multiplied by then we’ve got four multiplied by nine minus one multiplied by
negative one. And then, we’ve got plus nine
multiplied by negative eight multiplied by nine minus one multiplied by two. And we get that because we had
minus negative nine. And if we subtract a negative, it
turns positive.
And then, finally, you’ve got minus
six multiplied by negative eight multiplied by negative one minus four multiplied by
two. So, if we evaluate this, we’re
gonna get 37. And that’s because we had one
multiplied by then you’ve got 36 minus negative one, which is 36, add one, which is
37. Then, we’re gonna get minus
666. And that’s because we had nine
multiplied by negative 74. That’s cause we had negative 72
minus two, which is negative 74, which gives us negative 666.
And then, minus zero. And we get that because we have
negative eight multiplied by negative one, which is eight. Four multiplied by two, which is
eight. Eight minus eight is just zero. So, this gives us a final answer of
negative 629. So, therefore, we can say that if
we evaluate the determinant of the three-by-three matrix one, negative nine,
negative six, negative eight, four, one, two, negative one, nine, then the result is
negative 629.
