The Properties of determinat

The lecture covers section 5.1 from the textbook.
The determinant of a square matrix is a number that tells you quite a
bit about the matrix. Formally, it’s defined as:
det(A) = X?Sn
(?1)sgn()
n
Yi=1
ai(i).
where A = (aij). Well, that’s enlightening, isn’t it? That’s the last you’ll
see of that definition today, and instead we’ll talk about some properties
of the determinant that we’d like it to have. It turns out that with only a
few required properties, we can derive a whole bunch.
1 The Three Basic Properties
Recall that the inverse of a 2 × 2 matrix
A =  a b
c d 
is the matrix
1
A?1 =
1
ad ? bc  d ?b
?c a .
This is nice and well defined assuming ad ? bc 6= 0. If ad ? bc = 0 then
this formula doesn’t work, and in fact A is not invertible. For a 2×2matrix
the determinant is the number ad ? bc. We’ll be using the 2 × 2 example
repeatedly as we go over these properties.
The first basic property concerns the determinant of the identity matrix
I. In the 2 × 2 case the identity matrix is
I =  1 0
0 1 
and the determinant is 1 × 1 ? 0 × 0 = 1. In general we’d like for the
determinant of the n × n identity matrix
I =?
??
1
. . .
1
???
to be 1. In fact, this is the first defining property.
Next, we note that if we switch the rows of a 2 × 2 matrix:
 c d
a b 
the determinant is (bc?ad) = ?(ad?bc). So, the sign of the determinant
is switched. We want this property, that switching the rows changes the
sign of the determinant, to be our second defining property.
Finally, we note that the determinant of a 2×2matrix is a linear function
of each row separately. That is to say, if wemultiply a row, say the top row,
by a number t we get
2
 ta tb
c d 
which has determinant tad?tbc = t(ad?bc). So, multiplying a row by
tmultiplies the determinant by t. As for the other linearity property, when
we add two row vectors together we get
 a + a? b + b?
c d 
the determinant is ad ? bc + a?d