Linear Algebra

Section I. Definition 309
read aloud as “the sum, over all permutations φ, of terms having the form
t1,φ(1)t2,φ(2) ···tn,φ(n)|Pφ|”. This phrase is just a restating of the three-step
process (Step 1) for each permutation matrix, compute t1,φ(1)t2,φ(2) ···tn,φ(n) (Step 2) multiply that by |Pφ| and (Step 3) sum all such terms together.
3.10 Example The familiar formula for the determinant of a 2×2 matrix can
be derived in this way.
�
�
�
� t1,1
�
t1,2�
�
t2,1 t2,2�
= t1,1t2,2 ·|Pφ1 | + t1,2t2,1 ·|Pφ2 |
� �
�
= t1,1t2,2 · �1
0�
�
�0 1�
+ t1,2t2,1
� �
�
· �0
1�
�
�1 0�
= t1,1t2,2 − t1,2t2,1
(the second permutation matrix takes one row swap to pass to the identity).
Similarly, the formula for the determinant of a 3×3 matrix is this.
�
�t1,1
�
�t2,1
�
�t3,1
t1,2
t2,2
t3,2
�
t1,3�
�
t2,3�
�
t3,3�
= t1,1t2,2t3,3 |Pφ1 | + t1,1t2,3t3,2 |Pφ2 | + t1,2t2,1t3,3 |Pφ3 |
+ t1,2t2,3t3,1 |Pφ4 | + t1,3t2,1t3,2 |Pφ5 | + t1,3t2,2t3,1 |Pφ6 |
= t1,1t2,2t3,3 − t1,1t2,3t3,2 − t1,2t2,1t3,3
+ t1,2t2,3t3,1 + t1,3t2,1t3,2 − t1,3t2,2t3,1
Computing a determinant by permutation expansion usually takes longer
than Gauss’ method. However, here we are not trying to do the computation
efficiently, we are instead trying to give a determinant formula that we can
prove to be well-defined. While the permutation expansion is impractical for
computations, it is useful in proofs. In particular, we can use it for the result
that we are after.
3.11 Theorem For each n there is a n×n determinant function.
The proof is deferred to the following subsection. Also there is the proof of
the next result (they share some features).
3.12 Theorem The determinant of a matrix equals the determinant of its
transpose.
The consequence of this theorem is that, while we have so far stated results
in terms of rows (e.g., determinants are multilinear in their rows, row swaps
change the signum, etc.), all of the results also hold in terms of columns. The
final result gives examples.
3.13 Corollary A matrix with two equal columns is singular. Column swaps
change the sign of a determinant. Determinants are multilinear in their columns.
Proof. For the first statement, transposing the matrix results in a matrix with
the same determinant, and with two equal rows, and hence a determinant of
zero. The other two are proved in the same way. QED