Determinant of a Square Matrix

Determinant Definition
Let A = [aij ] be an n× n matrix (i.e., A is a square matrix). Given a pair of (i, j), we define Mij
to be the (n ? 1) × (n ? 1) matrix obtained by removing the i-th row and j-th column of A. For
example, suppose that
A = ?
?
1 2 1
3 0 -2
-1 -1 2
??
Then:
M21 =  2 1
-1 2 ,M22 =  1 1
-1 2 ,M32 =  1 1
3 -2 
We are now ready to define determinants:
Definition 1. Let A = [aij ] be an n × n matrix. If n = 1, its determinant, denoted as det(A),
equals a11. If n > 1, we first choose an arbitrary i ? [1, n], and then define the determinant of A
recursively as:
det(A) =
n
Xj=1
(?1)i+j · aij · det(Mij). (1)
Besides det(A), we may also denote the determinant of A as |A|. Henceforth, if we apply (1)
to compute det(A), we say that we expand A by row i. It is important to note that the value of
det(A) does not depend on the choice of i. We omit a proof of this fact, but illustrate it in the
following examples.
Example 1 (Second-Order Determinants). In general, if A = [aij ] is a 2 × 2 matrix, then
det (A) = a11a22 ? a12a21.
For instance:

2 1
-1 2
= 2 × 2 ? 1 × (?1) = 5.
We may verify the above by definition as follows. Choosing i = 1, we get:

2 1
-1 2
= (?1)1+1 · 2 · det(M11) + (?1)1+2 · 1 · det(M12)
= 2 × 2 + (?1) × (?1) = 5.
1
Alternatively, choosing i = 2, we get:

2 1
-1 2
= (?1)2+1 · (?1) · det(M21) + (?1)2+2 · 2 · det (M22)
= 1 × 1 + 2 × 2 = 5.
Example 2 (Third-Order Determinants). Suppose that
A = ?
?
1 2 1
3 0 -2
-1 -1 2
??
Choosing i = 1, we get:
det(A) = 1
0 ?2
?1 2 
? 2
3 ?2
?1 2 
+ 1
3 0
?1 ?1 = 1(0 ? 2) ? 2(6 ? 2) + 1(?3 ? 0) = ?13.
Alternatively, choosing i = 2, we get:
det(A) = ?3
2 1
?1 2
+ 0
1 1
?1 2
? (?2)
1 2
?1 ?1 = (?3)(4 + 1) + 0(2 + 1) + 2(?1 + 2) = ?13.
2 Properties of Determinants
Expansion by a Column. Definition 1 allows us to compute the determinant of a matrix by row
expansion. We may also achieve the same purpose by column expansion.
Lemma 1. Let A = [aij ] be an n × n matrix with n > 1. Choose an arbitrary j ? [1, n]. The
determinant of A equals:
det(A) =
n
Xi=1
(?1)i+j
· aij · det(Mij).
The value of the above equation does not depend on the choice of j.
We omit a proof but illustrate the lemma with an example below. Henceforth, if we compute
det(A) by the above lemma, we say that we expand A by column j.
Example 3. Suppose that
A = ?
?
1 2 1
3 0 -2
-1 -1 2
??
2