then det(A) =
ad - bc is found this way:
Determinants
The determinant of a square matrix is a real
number that gives us valuable information about the matrix. Its definition is
cumbersome, so it is in a special section at the end of this chapter. You will
see some of the uses of the determinant in later chapters. For now, let's find
out how to compute the determinant of a matrix so that we can use it later. The
symbols det(A) and |A| represent the determinant of A. In
this case, the straight bars do NOT mean absolute value; they represent the
determinant of the matrix. The determinant of a 1 by 1 matrix is simply the
element of the matrix. If A is the 2 by 2 matrix, then det(A) =
ad - bc is found this way:
You may remember ad-bc from the last chapter. If det(A)
≠ 0, then the inverse of the 2 by 2 matrix, A, is
,
which can also be written as
We have already found the determinant for a 2 by 2 matrix, so
let's look at the 3 by 3 matrix
.
In the 2 by 2 case, we subtracted products of the diagonals
from each other beginning with the main diagonal. If we do that with the 3 by 3
case, we will be leaving out 4 of the 9 numbers. Let's rewrite the first two
columns of A so that each number of the original matrix falls on a
diagonal. It looks like this:
If we only use the diagonals that have 3 numbers on them, we
will be using every number that was in our original matrix the same number of
times. The diagonals towards the upper left give us 6*2*0 = 0 and 1*3*2 = 6 and
3*4*4 = 48. If we add these together, we get 54. Now let's look at the diagonals
towards the upper right. They give us 1*2*4 = 8 and 3*4*0 = 0 and 6*4*2 = 48. If
we add these together, we get 56. Let's subtract the upper right from the upper
left like we did with the 2 by 2 case. We obtain | A| = -2. The method of
finding the determinant for the 3 by 3 matrix that we showed you is called
"repeat the columns" for obvious reasons. Look at the numbers you used and where
they came from. See if you can find a way to determine these same numbers
without writing down the columns each time. It could look like this:
| |