Hey guys, in this video I wanted to go over this proof problem.
It says let A be an n x n
idempotent matrix. Find all possible values of its determinant. This is a really simple proof.
It's basically just testing if you know basic determinant properties
and if you know what an idempotent matrix is. Since it's such a short proof before I actually go to
the proof, I wanted to go over some basic determinant properties
and we're going to use one of these properties in the proof. And this is not an exhaustive list of properties,
it's just some of the main ones.
I have if A is an n x n matrix,
then the determinant of k times A, where k is a constant
is equal to k to the N times the determinant of A, where N is
the size of the matrix. If it's 3x3 matrix, for example,
you'll have k to the third power. Then similarly if B is an n x n
matrix, then the determinant of B
times A, where this is matrix multiplication between two n x n
matrices, is equal to the scalar
multiplication of the determinant of B times the determinant of A. And for this,
the order doesn't matter because the determinant of B is a scalar and the determinant of A is a scalar.
You could write, for example, that this is equal to the determinant of A times the determinant of B
and that's still true. A nice property that you can get from
this property is that if you have the determinant of A times itself, which you can do because A is an n x n matrix.
This is equivalent to saying the determinant of A squared, which is going to be equal to the determinant of A
times the determinant of A, which is equal to the determinant of A
squared. We go from the determinant of A squared to the square of the determinant of A.
Similarly if you have an
invertible matrix, then the determinant of A inverse is equal to the inverse
of the determinant of A. And the last property is that the determinant of A transpose is always equal to the determinant
of A. For this problem
we're just going to use this second property here. And let me go ahead and solve the proof.
We have by assumption A is idempotent.
Thus we can say that A times A is equal to A squared is equal to A.
From this we can get the determinant of both sides. We have that the determinant of A
squared is equal to the determinant of A. From the determinant properties
we just went over, we know that the determinant of the square of a matrix is equal to the square of the
determinant of the matrix. This is equal to the determinant of A
squared and the right-hand side is still the determinant of A. We can move both to the left-hand side to get
the determinant of A
squared minus the determinant of A is equal to zero. We can factor out a
determinant of A from this. We get the determinant of A times the determinant of A
minus one is equal to zero. From this we can pull out the
determinant of A is equal to 0 or the determinant of A minus 1 is equal to 0.
And therefore we have that for any idempotent matrix, the determinant
has to be 0 or the determinant has to be 1. And this is our proof and we're done with this problem.
*chiptune music*
