This is a discussion of some of the aspects of determinants that seem to be overlooked (or under-emphasized)
in linear algebra classes and in pre-calculus classes covering matrixes. It is not a general discussion of
determinants.
1) Conceptually, What IS the Determinant?
The determinant of a matrix is a measure of the signedmagnitude of the matrix. The only
real strangeness is that the magnitude here can be signed, so determinants can be negative. I
don't know of a single word that means 'signed magnitude,' so we'll go with magnitude and keep in mind
that the value can be zero or negative.
What the magnitude means depends on the 'size' of the matrix, where the 'size' is
the number of rows (and 'size' is also the number of columns because for square matrices the number of
rows is equal to the number of columns).
For a 1×1 matrix the determinant is the value of the single element of
the matrix. You can think of this as the length of a vector from the origin
to a point on the number line specified by
the value of that single element.
For a 2×2 matrix the determinant is the signed area of a parallelogram
specified by the matrix. The two rows of the matrix specify two 2-dimensional
vectors and the parallelogram is specified by these two vectors. An example of
one of these appears below.
For a 3×3 matrix the determinant is the signed volume of a parallelpiped
specified by the matrix. This works the same way as the 2-dimensional case,
but in space rather than on a plane.
For an n×n matrix the determinant is the signed hyper-volume of an
n-dimensional parallelotope specified by the matrix.
The pattern should be obvious.
For a 1×1 matrix, the determinant is the signed length of the line specified by the 1-D vector.
For a 2×2 matrix, the determinant is the signed area of the parallelogram specified by the 2-D vectors.
For a 3×3 matrix, the determinant is the signed volume of the parallelpiped specified by the 3-D vectors.
2) Why Does Determinant Re-use the 'Absolute Value' Symbol?
Because absolute value is another form of magnitude, but for numbers rather than matrixes.
Instead of thinking of absolute value as 'dropping the negative sign,' think of it as 'distance from the origin.'
If you think of distance (or length) as a form of 'magnitude', then the two concepts are pretty close.
NOTE: You should be thinking of absolute value as 'distance from the origin' even if you aren't working with
matrixes. The absolute value of complex numbers is NOT computed by 'dropping the negative sign':
= 5
= 5
= 5
= 5
3) Where Does 'ad - bc' Come From?
A commonly used formula for calculating the determinant of a 2-dimensional
matrix is:
= ad - bc.
This is easy to compute, but seems fairly arbitrary. Or more than fairly arbitrary. Why does the formula work?
If we consider that a 2-dimensional determinant IS an area, we can draw the following
picture, replacing a, b, c, and d with x1, y1, x2, and y2 for clarity:
The determinant:
is the area of the grey parallelogram in the middle.
Capital letters 'A' - 'F' are the areas of the regions in which the letters reside.
The 'x1', 'x2', 'y1', and 'y2' values on the edges are lengths.
And now for some algebra.
Area of the entire rectangle is:
(x1 + x2) · (y1 + y2)
... and expanded is:
(x1·y1)
+
(x1·y2)
+
(x2·y1)
+
(x2·y2)
subtract piece 'A':
-
(x2·y1)
subtract piece 'B':
-
½(x1·y1)
subtract piece 'C':
-
½(x2·y2)
subtract piece 'D':
-
(x2·y1)
subtract piece 'E':
-
½(x1·y1)
subtract piece 'F':
-
½(x2·y2)
and ... the area of the Parallelogram:
(x1·y2)
-
(x2·y1)
So the formula IS just the area of the parallelogram ... with all the details of how to get
there omitted.
4) How Can Determinants Be Negative?
If the determinant is a magnitude, how can magnitudes be negative?
In some sense, magnitudes shouldn't be negative, but this is why the math folks have another name for this.
One way to think of the sign is that measuring the area while sweeping out the shape counter-clockwise results
in a positive measure of magnitude, while sweeping out the shape clockwise results in a negative magnitude. Thus
the following two matrixes have determinants with opposite signs.
For 2×2 matrixes the sweeping works the same way that the angles on a unit circle do:
counter-clockwise is positive, and clockwise is negative.
And Some (One!) Fairly Obvious Properties For 2×2 Matrixes
Identity Matrixes Result in Determinants That Are '1'
Identity matrixes are basically vectors of length '1' in each independent dimension.