Page 173 - 35Linear Algebra

8.1 The Determinant Formula 173

This is very cumbersome.
Luckily, it is very easy to compute the determinants of certain matrices.
i
For example, if M is diagonal, meaning that M = 0 whenever i 6= j, then
j
all summands of the determinant involving off-diagonal entries vanish and
X
n
2
1
det M = sgn(σ)m 1 σ(1) m 2 σ(2) · · · m n = m m · · · m .
1
2
σ(n)
n
σ
The determinant of a diagonal matrix is
the product of its diagonal entries.
Since the identity matrix is diagonal with all diagonal entries equal to one,
we have
det I = 1.
We would like to use the determinant to decide whether a matrix is in-
vertible. Previously, we computed the inverse of a matrix by applying row
operations. Therefore we ask what happens to the determinant when row
operations are applied to a matrix.
Swapping rows Lets swap rows i and j of a matrix M and then compute its determi-
nant. For the permutation σ, let ˆσ be the permutation obtained by swapping positions
i and j. Clearly
sgn(ˆσ) = −sgn(σ) .
0
Let M be the matrix M with rows i and j swapped. Then (assuming i < j):
det M 0 = X sgn(σ) m 1 · · · m j · · · m i · · · m n
σ(1) σ(i) σ(j) σ(n)
σ
X j
= sgn(σ) m 1 · · · m i · · · m · · · m n
σ(1) σ(j) σ(i) σ(n)
σ
X j
= (−sgn(ˆσ)) m 1 · · · m i · · · m · · · m n
ˆ σ(1) ˆ σ(i) ˆ σ(j) ˆ σ(n)
σ
X j
= − sgn(ˆσ) m 1 · · · m i · · · m · · · m n
ˆ σ(1) ˆ σ(i) ˆ σ(j) ˆ σ(n)
ˆ σ
= − det M.
P P
The step replacing by often causes confusion; it holds since we sum over all
σ ˆ σ
permutations (see review problem 3). Thus we see that swapping rows changes the
sign of the determinant. I.e.,
0
det M = − det M .

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